pax_global_header00006660000000000000000000000064125440460100014506gustar00rootroot0000000000000052 comment=23f01cbdf1496000fa6e5d0a2f6be91dd5c03cf6 pycode-browser-1.02/000077500000000000000000000000001254404601000143745ustar00rootroot00000000000000pycode-browser-1.02/Code/000077500000000000000000000000001254404601000152465ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/000077500000000000000000000000001254404601000163225ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/Arrays/000077500000000000000000000000001254404601000175635ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/Arrays/cross.py000077500000000000000000000001441254404601000212700ustar00rootroot00000000000000#Example cross.py from numpy import * a = array([1,2,3]) b = array([4,5,6]) c = cross(a,b) print c pycode-browser-1.02/Code/Maths/Arrays/fileio.py000077500000000000000000000001611254404601000214050ustar00rootroot00000000000000#Example fileio.py from numpy import * a = arange(10) a.tofile('myfile.dat') b = fromfile('myfile.dat') print b pycode-browser-1.02/Code/Maths/Arrays/inv.py000077500000000000000000000002231254404601000207310ustar00rootroot00000000000000#Example inv.py from numpy import * a = array([ [4,1,-2], [2,-3,3], [-6,-2,1] ], dtype='float') ainv = linalg.inv(a) print ainv print dot(a,ainv) pycode-browser-1.02/Code/Maths/Arrays/numpy1.py000077500000000000000000000001461254404601000213720ustar00rootroot00000000000000#Example numpy1.py from numpy import * x = array( [1,2] ) # Make array from list print x , type(x) pycode-browser-1.02/Code/Maths/Arrays/numpy2.py000077500000000000000000000003131254404601000213670ustar00rootroot00000000000000#Example numpy2.py from numpy import * a = arange(1.0, 2.0, 0.1) # start, stop & step print a b = linspace(1,2,11) print b c = ones(5) print c d = zeros(5) print d e = random.rand(5) print e pycode-browser-1.02/Code/Maths/Arrays/numpy3.py000077500000000000000000000002071254404601000213720ustar00rootroot00000000000000#Example numpy3.py from numpy import * a = [ [1,2] , [3,4] ] # make a list of lists x = array(a) # and convert to an array print a pycode-browser-1.02/Code/Maths/Arrays/numpy4.py000077500000000000000000000001611254404601000213720ustar00rootroot00000000000000#Example numpy4.py from numpy import * a = arange(10) print a a = a.reshape(5,2) # 5 rows and 2 columns print a pycode-browser-1.02/Code/Maths/Arrays/numpy5.py000077500000000000000000000002201254404601000213670ustar00rootroot00000000000000#Example numpy5.py from numpy import * a = array([ [1.0, 2.0] , [3.0, 4.0] ]) # make a numpy array print a print a * 5 print a * a print a / a pycode-browser-1.02/Code/Maths/Arrays/oper.py000077500000000000000000000001621254404601000211040ustar00rootroot00000000000000#Example oper.py from numpy import * a = array([[2,3], [4,5]]) b = array([[1,2], [3,0]]) print a + b print a * b pycode-browser-1.02/Code/Maths/Arrays/reshape.py000077500000000000000000000001261254404601000215660ustar00rootroot00000000000000#Example reshape.py from numpy import * a = arange(20) b = reshape(a, [4,5]) print b pycode-browser-1.02/Code/Maths/Calculus/000077500000000000000000000000001254404601000200755ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/Calculus/compareEuRK4.py000077500000000000000000000011631254404601000227140ustar00rootroot00000000000000''' Sine function is calculated using its derivative, cosine. Using Euler and RK4 method, with ''' import math def rk4(x, y, yprime, dx = 0.01): # x, y , derivative, stepsize k1 = dx * yprime(x) k2 = dx * yprime(x + dx/2.0) k3 = dx * yprime(x + dx/2.0) k4 = dx * yprime(x + dx) return y + ( k1/6 + k2/3 + k3/3 + k4/6 ) h = 0.01 # stepsize x = 0.0 # initail values y = 0.0 # for Euler z = 0.0 # for RK4 while x < math.pi: print x, y - math.sin(x), z - math.sin(x) # errors y = y + h * math.cos(x) # Euler method z = rk4(x,z,math.cos,h) # Runge-Kutta method x = x + h pycode-browser-1.02/Code/Maths/Calculus/diff.py000077500000000000000000000002771254404601000213700ustar00rootroot00000000000000#Example diff.py def f(x): return x**3 def deriv(x,dx=0.005): df = f(x+dx/2)-f(x-dx/2) return df/dx print deriv(2.0) print deriv(2.0, 0.1) print deriv(2.0, 0.0001) pycode-browser-1.02/Code/Maths/Calculus/diff1.py000077500000000000000000000007421254404601000214460ustar00rootroot00000000000000from pylab import * dx = 0.1 # value of x increment x = arange(0,10, dx) y = sin(x) yprime = [] # empty list for k in range(99): # from 100 points we get 99 diference values dy = y[k+1]-y[k] yprime.append(dy/dx) x1 = x[:-1] # A new array without the last element x1 = x1 + dx/2 # The derivative corresponds to the middle point plot(x1, yprime, '+') plot(x1, cos(x1)) # Cross check with the analitic value show() pycode-browser-1.02/Code/Maths/Calculus/euler.py000077500000000000000000000002731254404601000215700ustar00rootroot00000000000000import math h = 0.01 # stepsize x = 0.0 # initail values y = 0.0 while x < math.pi: print x, y, math.sin(x) y = y + h * math.cos(x) # Euler method x = x + h pycode-browser-1.02/Code/Maths/Calculus/fit1.py000077500000000000000000000007371254404601000213240ustar00rootroot00000000000000from pylab import * from scipy import * from scipy.optimize import leastsq def err_func(p,y,x): A,k,theta = p return y - A*sin(2*pi*k*x+theta) def evaluate(x,p): return p[0] * sin(2*pi*p[1]*x+p[2]) ax = arange(0, 0.2, 0.001) A,k,theta = 5, 50.0, pi/3 y_true = A*sin(2*pi*k*ax+theta) y_meas = y_true + 0.2*randn(len(ax)) p0 = [6, 50.0, pi/3] plsq = leastsq(err_func,p0,args=(y_meas,ax)) print plsq plot(ax,y_true) plot(ax,y_meas,'o') plot(ax,evaluate(ax,plsq[0])) show() pycode-browser-1.02/Code/Maths/Calculus/fit2.py000077500000000000000000000007371254404601000213250ustar00rootroot00000000000000from pylab import * from scipy import * from scipy.optimize import leastsq def err_func(p,y,x): A,k,theta = p return y - A*sin(2*pi*k*x+theta) def evaluate(x,p): return p[0] * sin(2*pi*p[1]*x+p[2]) ax = arange(0, 0.2, 0.001) A,k,theta = 5, 50.0, pi/3 y_true = A*sin(2*pi*k*ax+theta) y_meas = y_true + 0.2*randn(len(ax)) p0 = [6, 50.0, pi/3] plsq = leastsq(err_func,p0,args=(y_meas,ax)) print plsq plot(ax,y_true) plot(ax,y_meas,'o') plot(ax,evaluate(ax,plsq[0])) show() pycode-browser-1.02/Code/Maths/Calculus/rk4.py000077500000000000000000000010401254404601000211450ustar00rootroot00000000000000''' Solving initial value problem using 4th order Runge-Kutta method. Sine function is calculated using its derivative, cosine. ''' import math def rk4(x, y, yprime, dx = 0.01): # x, y , derivative, stepsize k1 = dx * yprime(x) k2 = dx * yprime(x + dx/2.0) k3 = dx * yprime(x + dx/2.0) k4 = dx * yprime(x + dx) return y + ( k1/6 + k2/3 + k3/3 + k4/6 ) h = 0.01 # stepsize x = 0.0 # initail values y = 0.0 while x < math.pi: print x, y, math.sin(x) y = rk4(x,y,math.cos) # Runge-Kutta method x = x + h pycode-browser-1.02/Code/Maths/Calculus/rk_sin.py000077500000000000000000000011311254404601000217330ustar00rootroot00000000000000from pylab import * def func(t): return cos(t) def rk4(s,t): k1 = dt * func(t) k2 = dt * func(t + dt/2.0) k3 = dt * func(t + dt/2.0) k4 = dt * func(t + dt) return s + ( k1/6 + k2/3 + k3/3 + k4/6 ) t = 0.0 # Stating angle dt = 0.01 # value of angle increment val = 0.0 # value of the 'sine' function at t = 0 tm = [0.0] # List to store theta values res = [0.0] # RK4 results while t < 2*pi: val = rk4(val,t) # get the new value t = t + dt tm.append(t) res.append(val) plot(tm, res,'+') plot(tm, sin(tm)) # compare with actual curve show() pycode-browser-1.02/Code/Maths/Calculus/trapez.py000077500000000000000000000004471254404601000217640ustar00rootroot00000000000000from math import * def sqr(a): return a**2 def trapez(f, a, b, n): h = (b-a) / n sum = f(a) for i in range (1,n): sum = sum + 2 * f(a + h * i) sum = sum + f(b) return 0.5 * h * sum print trapez(sin,0.,pi,100) print trapez(sqr,0.,2.,100) print trapez(sqr,0,2,100) # Why the error ? pycode-browser-1.02/Code/Maths/Calculus/vdiff.py000077500000000000000000000004071254404601000215510ustar00rootroot00000000000000#Example vdiff.py from pylab import * def f(x): return sin(x) def deriv(x,dx=0.005): df = f(x+dx/2)-f(x-dx/2) return df/dx vecderiv = vectorize(deriv) x = linspace(-2*pi, 2*pi, 200) y = vecderiv(x) plot(x,y,'+') plot(x,cos(x)) show() pycode-browser-1.02/Code/Maths/FamousCurves/000077500000000000000000000000001254404601000207445ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/FamousCurves/archi.py000077500000000000000000000001731254404601000224100ustar00rootroot00000000000000#Example archi.py from pylab import * a = 2 th= linspace(0, 10*pi,200) r = a*th polar(th,r) axis([0, 2*pi, 0, 70]) show() pycode-browser-1.02/Code/Maths/FamousCurves/archi.txt000077500000000000000000000016121254404601000225760ustar00rootroot00000000000000This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file.This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file.This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file.This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file.This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file. pycode-browser-1.02/Code/Maths/FamousCurves/astro.py000077500000000000000000000001751254404601000224540ustar00rootroot00000000000000#Example astro.py from pylab import * a = 2 x = linspace(0,a,100) y = ( a**(2.0/3) - x**(2.0/3) )**(3.0/2) plot(x,y) show() pycode-browser-1.02/Code/Maths/FamousCurves/astropar.py000077500000000000000000000002001254404601000231440ustar00rootroot00000000000000#Example astropar.py from pylab import * a = 2 t = linspace(-2*a,2*a,101) x = a * cos(t)**3 y = a * sin(t)**3 plot(x,y) show() pycode-browser-1.02/Code/Maths/FamousCurves/cover.py000077500000000000000000000001761254404601000224430ustar00rootroot00000000000000from pylab import * th = linspace(0, 10*pi,1000) r = 4* sin(8*th) polar(th,r) r = sqrt(th) polar(th,r) polar(th, -r) show() pycode-browser-1.02/Code/Maths/FamousCurves/ellipse.py000077500000000000000000000002101254404601000227470ustar00rootroot00000000000000#Example ellipse.py from pylab import * a = 2 b = 3 t = linspace(0, 2 * pi, 100) x = a * sin(t) y = b * cos(t) plot(x,y) show() pycode-browser-1.02/Code/Maths/FamousCurves/fermat.py000077500000000000000000000001761254404601000226030ustar00rootroot00000000000000#Example fermat.py from pylab import * a = 2 th= linspace(0, 10*pi,200) r = sqrt(a**2 * th) polar(th,r) polar(th, -r) show() pycode-browser-1.02/Code/Maths/FamousCurves/lissa.py000077500000000000000000000002611254404601000224330ustar00rootroot00000000000000#Example lissa.py from pylab import * a = 2 b = 3 t= linspace(0, 2*pi,100) x = a * sin(2*t) y = b * cos(t) plot(x,y) x = a * sin(3*t) y = b * cos(2*t) plot(x,y) show() pycode-browser-1.02/Code/Maths/Functions/000077500000000000000000000000001254404601000202725ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/Functions/bes1.py000077500000000000000000000003231254404601000214770ustar00rootroot00000000000000from pylab import * from scipy import special def vj0(xarray): y = [] for x in xarray: val = special.j0(x) # Compute Jo y.append(val) return y a = linspace(0,10,100) b = vj0(a) plot(a,b) show() pycode-browser-1.02/Code/Maths/Functions/bes2.py000077500000000000000000000003531254404601000215030ustar00rootroot00000000000000from pylab import * from scipy import * def vjn(n,xarray): y = [] for x in xarray: val = special.jn(n,x) # Compute Jn(x) y.append(val) return y a = linspace(0,10,100) for n in range(5): b = vjn(n,a) plot(a,b) show() pycode-browser-1.02/Code/Maths/Functions/bes3.py000077500000000000000000000011301254404601000214760ustar00rootroot00000000000000from scipy import * from scipy import special import pylab def jn(n,x): jn = 0.0 for k in range(30): num = (-1)**k * (x/2)**(2*k) den = factorial(k)*factorial(n+k) jn = jn + num/den return jn * (x/2)**n def vjn_local(n,xarray): y = [] for x in xarray: val = jn(n,x) # Jn(x) using our function y.append(val) return y def vjn(n,xarray): y = [] for x in xarray: val = special.jn(n,x) # Compute Jn(x) y.append(val) return y a = linspace(0,10,100) for n in range(2): b = vjn(n,a) c = vjn_local(n,a) pylab.plot(a,b) pylab.plot(a,c,marker = '+') pylab.show() pycode-browser-1.02/Code/Maths/Functions/legen1.py000077500000000000000000000002441254404601000220220ustar00rootroot00000000000000from scipy import linspace, special import pylab x = linspace(-1,1,100) for n in range(1,6): leg = special.legendre(n) y = leg(x) pylab.plot(x,y) pylab.show() pycode-browser-1.02/Code/Maths/Graphs/000077500000000000000000000000001254404601000175465ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/Graphs/arcs.py000077500000000000000000000002511254404601000210510ustar00rootroot00000000000000#Example arcs.py from pylab import * a = 10.0 for a in range(5,21,5): th = linspace(0, pi * a/10, 200) x = a * cos(th) y = a * sin(th) plot(x,y) show() pycode-browser-1.02/Code/Maths/Graphs/circ.py000077500000000000000000000002451254404601000210440ustar00rootroot00000000000000#Example circ.py from pylab import * a = 10.0 x = linspace(-a, a , 200) yupper = sqrt(a**2 - x**2) ylower = -sqrt(a**2 - x**2) plot(x,yupper) plot(x,ylower) show() pycode-browser-1.02/Code/Maths/Graphs/circpar.py000077500000000000000000000001771254404601000215530ustar00rootroot00000000000000#Example circpar.py from pylab import * a = 10.0 th = linspace(0, 2*pi, 200) x = a * cos(th) y = a * sin(th) plot(x,y) show() pycode-browser-1.02/Code/Maths/Graphs/julia.py000077500000000000000000000021061254404601000212260ustar00rootroot00000000000000''' Region of a complex plane ranging from -1 to +1 in both real and imaginary axes is rpresented using a 2 dimensional matrix having X x Y elements.For X and Y equal to 200, the stepsize in the complex plane is 2.0/200 = 0.01. The nature of the pattern depends very much on the value of c. ''' from pylab import * X = 200 Y = 200 rlim = 1.0 ilim = 1.0 rscale = 2*rlim / X iscale = 2*ilim / Y MAXIT = 100 MAXABS = 2.0 c = 0.02 - 0.8j # The constant in equation z**2 + c m = zeros([X,Y],dtype=uint8) # A two dimensional array def numit(x,y): # number of iterations to diverge z = complex(x,y) for k in range(MAXIT): if abs(z) <= MAXABS: z = z**2 + c else: return k # diverged after k trials return MAXIT # did not diverge, for x in range(X): for y in range(Y): re = rscale * x - rlim # complex number represented im = iscale * y - ilim # by the (x,y) coordinate m[x][y] = numit(re,im) # get the color for (x,y) imshow(m) # Colored plot using the two dimensional matrix show() pycode-browser-1.02/Code/Maths/Graphs/npsin.py000077500000000000000000000002001254404601000212420ustar00rootroot00000000000000#Example npsin.py from pylab import * x = linspace(-pi, pi , 200) y = sin(x) y1 = sin(x*x) plot(x,y) plot(x,y1,'r') show() pycode-browser-1.02/Code/Maths/Graphs/plot1.py000077500000000000000000000000651254404601000211630ustar00rootroot00000000000000from pylab import * x = range(10) plot(x) show() pycode-browser-1.02/Code/Maths/Graphs/plot2.py000077500000000000000000000001721254404601000211630ustar00rootroot00000000000000from pylab import * x = range(10,20) y = range(40,50) plot(x,y, marker = '+', color = 'red', markersize = 10) show() pycode-browser-1.02/Code/Maths/Graphs/plot3.py000077500000000000000000000003021254404601000211570ustar00rootroot00000000000000from pylab import * x = [] # empty lists y = [] for k in range(100): ang = 0.1 * k sang = sin(ang) x.append(ang) y.append(sang) plot(x,y) show() pycode-browser-1.02/Code/Maths/Graphs/plot4.py000077500000000000000000000002061254404601000211630ustar00rootroot00000000000000#Example plot4.py from pylab import * t = arange(0.0, 5.0, 0.2) plot(t, t**2,'x') # t^{2} plot(t, t**3,'ro') # t^{3} show() pycode-browser-1.02/Code/Maths/Graphs/polar.py000077500000000000000000000002131254404601000212340ustar00rootroot00000000000000#Example polar.py from pylab import * th = linspace(0,2*pi,100) r = 5 * ones(100) # radius = 5 axis([0, 2*pi, 0, 10]) polar(th,r) show() pycode-browser-1.02/Code/Maths/Graphs/sine.py000077500000000000000000000001461254404601000210620ustar00rootroot00000000000000#Example sine.py import math x = 0.0 while x < math.pi * 4: print x , math.sin(x) x = x + 0.1pycode-browser-1.02/Code/Maths/Polynomials/000077500000000000000000000000001254404601000206305ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/Polynomials/poly.py000077500000000000000000000002601254404601000221660ustar00rootroot00000000000000#Example poly.py from pylab import * a = poly1d([3,4,5]) b = poly1d([6,7]) c = a * b + 5 d = c/a print a print b print a * b print d[0], d[1] print a.deriv() print a.integ() pycode-browser-1.02/Code/Maths/Polynomials/poly1.py000077500000000000000000000004431254404601000222520ustar00rootroot00000000000000import scipy import pylab a = scipy.poly1d([3,4,5]) print a, ' is the polynomial' print a*a, 'is its square' print a.deriv(), ' is its derivative' print a.integ(), ' is its integral' print a(0.5), 'is its value at x = 0.5' x = scipy.linspace(0,5,100) b = a(x) pylab.plot(x,b) pylab.show() pycode-browser-1.02/Code/Maths/Polynomials/polyplot.py000077500000000000000000000002201254404601000230610ustar00rootroot00000000000000#Example polyplot.py from pylab import * x = linspace(-pi, pi, 100) a = poly1d([-1.0/5040,0,1.0/120,0,-1.0/6,0,1,0]) y = a(x) plot(x,y) show() pycode-browser-1.02/Code/Maths/Series/000077500000000000000000000000001254404601000175545ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/Series/fourier_sawtooth.py000077500000000000000000000003271254404601000235360ustar00rootroot00000000000000#Example fourier_sawtooth.py from pylab import * N = 100 # number of points x = linspace(-pi, pi, N) y = zeros(N) for n in range(1,10): term = (-1)**(n+1) * sin(n*x) / n y = y + term plot(x,y) show() pycode-browser-1.02/Code/Maths/Series/fourier_square.py000077500000000000000000000003271254404601000231660ustar00rootroot00000000000000#Example fourier_square.py from pylab import * N = 100 # number of points x = linspace(0.0, 2 * pi, N) y = zeros(N) for n in range(5): term = sin((2*n+1)*x) / (2*n+1) y = y + term plot(x,y) show() pycode-browser-1.02/Code/Maths/Series/series_cos.py000077500000000000000000000005041254404601000222660ustar00rootroot00000000000000import scipy, pylab def mycos(x): res = 0.0 for n in range(18): res = res + (-1)**n * (x ** (2*n)) / scipy.factorial(2*n) return res def vmycos(ax): y = [] for x in ax: y.append(mycos(x)) return y x = scipy.linspace(0,4*scipy.pi,100) y = vmycos(x) pylab.plot(x,y) pylab.plot(x,scipy.cos(x),'+') pylab.show() pycode-browser-1.02/Code/Maths/Series/series_sin.py000077500000000000000000000004631254404601000222770ustar00rootroot00000000000000#Example series_sin.py from pylab import * from scipy import factorial x = linspace(-pi, pi, 50) y = zeros(50) for n in range(5): term = (-1)**(n) * (x**(2*n+1)) / factorial(2*n+1) y = y + term plot(x,term) plot(x, y, '+b') plot(x, sin(x),'xr') # compare with the real one show() pycode-browser-1.02/Code/Maths/archi.py000077500000000000000000000001731254404601000177660ustar00rootroot00000000000000#Example archi.py from pylab import * a = 2 th= linspace(0, 10*pi,200) r = a*th polar(th,r) axis([0, 2*pi, 0, 70]) show() pycode-browser-1.02/Code/Maths/archi.txt000077500000000000000000000016121254404601000201540ustar00rootroot00000000000000This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file.This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file.This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file.This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file.This is the description of archi.py file. You can add description to any python file by adding a .txt file with the description. The file name should be the same as the python file. pycode-browser-1.02/Code/Maths/arcs.py000077500000000000000000000002511254404601000176250ustar00rootroot00000000000000#Example arcs.py from pylab import * a = 10.0 for a in range(5,21,5): th = linspace(0, pi * a/10, 200) x = a * cos(th) y = a * sin(th) plot(x,y) show() pycode-browser-1.02/Code/Maths/area.py000077500000000000000000000001601254404601000176040ustar00rootroot00000000000000#Example: area.py pi = 3.1416 r = input('Enter Radius ') a = pi * r ** 2 #A=\pi r^{2} print 'Area = ', a pycode-browser-1.02/Code/Maths/astro.py000077500000000000000000000001751254404601000200320ustar00rootroot00000000000000#Example astro.py from pylab import * a = 2 x = linspace(0,a,100) y = ( a**(2.0/3) - x**(2.0/3) )**(3.0/2) plot(x,y) show() pycode-browser-1.02/Code/Maths/astropar.py000077500000000000000000000002001254404601000205220ustar00rootroot00000000000000#Example astropar.py from pylab import * a = 2 t = linspace(-2*a,2*a,101) x = a * cos(t)**3 y = a * sin(t)**3 plot(x,y) show() pycode-browser-1.02/Code/Maths/badtable.py000077500000000000000000000001231254404601000204310ustar00rootroot00000000000000#Example: badtable.py print 1 * 8 print 2 * 8 print 3 * 8 print 4 * 8 print 5 * 8 pycode-browser-1.02/Code/Maths/big.py000077500000000000000000000002331254404601000174360ustar00rootroot00000000000000#Example: big.py x = input('Enter a number ') if x > 10: print 'Bigger Number' elif x < 10: print 'Smaller Number' else: print 'Same Number' pycode-browser-1.02/Code/Maths/big2.py000077500000000000000000000001761254404601000175260ustar00rootroot00000000000000#Example: big2.py x = 1 while x < 11: if x < 5: print 'Samll ', x else: print 'Big ', x print 'Done' pycode-browser-1.02/Code/Maths/big3.py000077500000000000000000000002061254404601000175210ustar00rootroot00000000000000#Example: big3.py x = 1 while x < 100: print x if x > 10: print 'Enough of this' break x = x + 1 print 'Done' pycode-browser-1.02/Code/Maths/circ.py000077500000000000000000000002451254404601000176200ustar00rootroot00000000000000#Example circ.py from pylab import * a = 10.0 x = linspace(-a, a , 200) yupper = sqrt(a**2 - x**2) ylower = -sqrt(a**2 - x**2) plot(x,yupper) plot(x,ylower) show() pycode-browser-1.02/Code/Maths/circpar.py000077500000000000000000000001771254404601000203270ustar00rootroot00000000000000#Example circpar.py from pylab import * a = 10.0 th = linspace(0, 2*pi, 200) x = a * cos(th) y = a * sin(th) plot(x,y) show() pycode-browser-1.02/Code/Maths/cloth.py000077500000000000000000000004701254404601000200110ustar00rootroot00000000000000class clothing: def __init__(self,colour, rate): self.colour = colour self.rate = rate def getcolour(self): return self.colour cheapblue = clothing('blue', 20.0) costlyred = clothing('red', 100.0) print cheapblue.rate print costlyredpycode-browser-1.02/Code/Maths/cloth2.py000077500000000000000000000005741254404601000201000ustar00rootroot00000000000000class clothing: def __init__(self,colour, rate): self.colour = colour self.rate = rate def getcolour(self): return self.colour def __str__(self): return '%s Cloth at Rs. %6.2f per sqmtr'%(self.colour, self.rate) cheapblue = clothing('blue', 20.0) costlyred = clothing('red', 100.0) print cheapblue.rate print costlyredpycode-browser-1.02/Code/Maths/clothbill.py000077500000000000000000000022241254404601000206530ustar00rootroot00000000000000class clothing: def __init__(self,colour, rate): self.colour = colour self.rate = rate def getcolour(self): return self.colour def __str__(self): return '%s Cloth at Rs. %6.2f per sqmtr'%(self.colour, self.rate) class pants(clothing): def __init__(self, cloth, size, labour): self.labour = labour self.size = size clothing.__init__(self,cloth.colour, cloth.rate) def __str__(self): return '%-10s Pants'%(self.colour) def getcost(self): return self.rate * self.size + self.labour class shirt(clothing): def __init__(self, cloth, size, labour): self.labour = labour self.size = size clothing.__init__(self,cloth.colour, cloth.rate) def __str__(self): return '%-10s Shirt'%(self.colour) def getcost(self): return self.rate * self.size + self.labour cheapblue = clothing('blue', 20.0) costlyred = clothing('red', 100.0) items = [] items.append(pants(cheapblue, 2.0, 150.0) ) items.append(shirt(costlyred, 1.5, 130.0) ) for k in items: print k, k.getcost() pycode-browser-1.02/Code/Maths/compareEuRK4.py000077500000000000000000000011631254404601000211410ustar00rootroot00000000000000''' Sine function is calculated using its derivative, cosine. Using Euler and RK4 method, with ''' import math def rk4(x, y, yprime, dx = 0.01): # x, y , derivative, stepsize k1 = dx * yprime(x) k2 = dx * yprime(x + dx/2.0) k3 = dx * yprime(x + dx/2.0) k4 = dx * yprime(x + dx) return y + ( k1/6 + k2/3 + k3/3 + k4/6 ) h = 0.01 # stepsize x = 0.0 # initail values y = 0.0 # for Euler z = 0.0 # for RK4 while x < math.pi: print x, y - math.sin(x), z - math.sin(x) # errors y = y + h * math.cos(x) # Euler method z = rk4(x,z,math.cos,h) # Runge-Kutta method x = x + h pycode-browser-1.02/Code/Maths/conflict.py000077500000000000000000000003651254404601000205040ustar00rootroot00000000000000#Example conflict.py from numpy import * x = linspace(0, 5, 10) # make a 10 element array print sin(x) # sin of scipy can handle arrays from math import * # sin of math will be called now print sin(x) # will give ERROR pycode-browser-1.02/Code/Maths/cover.py000077500000000000000000000001761254404601000200210ustar00rootroot00000000000000from pylab import * th = linspace(0, 10*pi,1000) r = 4* sin(8*th) polar(th,r) r = sqrt(th) polar(th,r) polar(th, -r) show() pycode-browser-1.02/Code/Maths/cross.py000077500000000000000000000001441254404601000200270ustar00rootroot00000000000000#Example cross.py from numpy import * a = array([1,2,3]) b = array([4,5,6]) c = cross(a,b) print c pycode-browser-1.02/Code/Maths/diff.py000077500000000000000000000002771254404601000176150ustar00rootroot00000000000000#Example diff.py def f(x): return x**3 def deriv(x,dx=0.005): df = f(x+dx/2)-f(x-dx/2) return df/dx print deriv(2.0) print deriv(2.0, 0.1) print deriv(2.0, 0.0001) pycode-browser-1.02/Code/Maths/ellipse.py000077500000000000000000000002101254404601000203250ustar00rootroot00000000000000#Example ellipse.py from pylab import * a = 2 b = 3 t = linspace(0, 2 * pi, 100) x = a * sin(t) y = b * cos(t) plot(x,y) show() pycode-browser-1.02/Code/Maths/euler.py000077500000000000000000000002731254404601000200150ustar00rootroot00000000000000import math h = 0.01 # stepsize x = 0.0 # initail values y = 0.0 while x < math.pi: print x, y, math.sin(x) y = y + h * math.cos(x) # Euler method x = x + h pycode-browser-1.02/Code/Maths/factor.py000077500000000000000000000002441254404601000201550ustar00rootroot00000000000000#Example factor.py def factorial(n): # a recursive function if n == 0: return 1 else: return n * factorial(n-1) print factorial(10) pycode-browser-1.02/Code/Maths/fermat.py000077500000000000000000000001761254404601000201610ustar00rootroot00000000000000#Example fermat.py from pylab import * a = 2 th= linspace(0, 10*pi,200) r = sqrt(a**2 * th) polar(th,r) polar(th, -r) show() pycode-browser-1.02/Code/Maths/fileio.py000077500000000000000000000001611254404601000201440ustar00rootroot00000000000000#Example fileio.py from numpy import * a = arange(10) a.tofile('myfile.dat') b = fromfile('myfile.dat') print b pycode-browser-1.02/Code/Maths/filelist.html000077500000000000000000000027111254404601000210270ustar00rootroot00000000000000 filelist archi.py
arcs.py
area.py
astro.py
astropar.py
badtable.py
big.py
big2.py
big3.py
circ.py
circpar.py
cloth.py  
cloth2.py
clothbill.py
compareEuRK4.py
conflict.py
cover.py   
cross.py
diff.py
ellipse.py
euler.py  
factor.py
fermat.py
fileio.py 
first.py  
forloop.py
forloop2.py
forloop3.py
forloop4.py
format.py 
format2.py
fourier_sawtooth.py
fourier_square.py 
func.py
inv.py
julia.py
kin.py 
kin2.py
lissa.py
mathlocal.py
mathlocal2.py
mathsin.py  
mathsin2.py 
max.py
npsin.py
numpy1.py
numpy2.py
numpy3.py
numpy4.py
numpy5.py
oper.py 
pants.py
pickledump.py
pickleload.py
plot1.py
plot2.py
plot3.py
plot4.py
polar.py
poly.py
polyplot.py
power.py  
reshape.py
rfile.py  
rfile2.py 
rfile3.py 
rk4.py    
second.py 
series_sin.py
sine.py
table.py
test.pck
third.py
trapez.py
vdiff.py
wfile.py
wfile2.py

pycode-browser-1.02/Code/Maths/first.py000077500000000000000000000005441254404601000200310ustar00rootroot00000000000000 #Example: first.py ''' This is Python program starting with a multi-line comment, enclosed within triple quotes. In a single line, anything after a # sign is a comment ''' x = 10 print x, type(x) #print x and its type y = 10.4 print y, type(y) z = 3 + 4j print z, type(z) s1 = 'I am a String ' s2 = 'me too' print s1, s2, type(s1) pycode-browser-1.02/Code/Maths/forloop.py000077500000000000000000000001741254404601000203610ustar00rootroot00000000000000#Example: forloop.py a = 'my name' for ch in a: print ch b = ['hello', 3.4, 2345, 3+5j] for item in b: print item pycode-browser-1.02/Code/Maths/forloop2.py000077500000000000000000000001321254404601000204350ustar00rootroot00000000000000#Example: forloop2.py mylist = range(5) print mylist for item in mylist: print item pycode-browser-1.02/Code/Maths/forloop3.py000077500000000000000000000001211254404601000204340ustar00rootroot00000000000000#Example: forloop3.py mylist = range(5,51,5) for item in mylist: print item pycode-browser-1.02/Code/Maths/forloop4.py000077500000000000000000000001421254404601000204400ustar00rootroot00000000000000a = [2, 5, -3, 4, -2, 12] size = len(a) for k in range(size): if a[k] < 0: a[k] = 0 print a pycode-browser-1.02/Code/Maths/format.py000077500000000000000000000001301254404601000201610ustar00rootroot00000000000000#Example: format.py a = 2.0 /3 print a print 'a = %5.3f' %(a) # upto 3 decimal places pycode-browser-1.02/Code/Maths/format2.py000077500000000000000000000003141254404601000202470ustar00rootroot00000000000000#Example: format2.py a = 'justify as you like' print '%30s'%a print '%-30s'%a # minus sign for left justification for k in range(1,11): # A good looking table print '5 x %2d = %2d' %(k, k*5)pycode-browser-1.02/Code/Maths/fourier_sawtooth.py000077500000000000000000000003271254404601000223040ustar00rootroot00000000000000#Example fourier_sawtooth.py from pylab import * N = 100 # number of points x = linspace(-pi, pi, N) y = zeros(N) for n in range(1,10): term = (-1)**(n+1) * sin(n*x) / n y = y + term plot(x,y) show() pycode-browser-1.02/Code/Maths/fourier_square.py000077500000000000000000000003271254404601000217340ustar00rootroot00000000000000#Example fourier_square.py from pylab import * N = 100 # number of points x = linspace(0.0, 2 * pi, N) y = zeros(N) for n in range(5): term = sin((2*n+1)*x) / (2*n+1) y = y + term plot(x,y) show() pycode-browser-1.02/Code/Maths/func.py000077500000000000000000000001321254404601000176260ustar00rootroot00000000000000#Example func.py def sum(a,b): # a trivial function return a + b print sum(3, 4) pycode-browser-1.02/Code/Maths/inv.py000077500000000000000000000002231254404601000174700ustar00rootroot00000000000000#Example inv.py from numpy import * a = array([ [4,1,-2], [2,-3,3], [-6,-2,1] ], dtype='float') ainv = linalg.inv(a) print ainv print dot(a,ainv) pycode-browser-1.02/Code/Maths/julia.py000077500000000000000000000021061254404601000200020ustar00rootroot00000000000000''' Region of a complex plane ranging from -1 to +1 in both real and imaginary axes is rpresented using a 2 dimensional matrix having X x Y elements.For X and Y equal to 200, the stepsize in the complex plane is 2.0/200 = 0.01. The nature of the pattern depends very much on the value of c. ''' from pylab import * X = 200 Y = 200 rlim = 1.0 ilim = 1.0 rscale = 2*rlim / X iscale = 2*ilim / Y MAXIT = 100 MAXABS = 2.0 c = 0.02 - 0.8j # The constant in equation z**2 + c m = zeros([X,Y],dtype=uint8) # A two dimensional array def numit(x,y): # number of iterations to diverge z = complex(x,y) for k in range(MAXIT): if abs(z) <= MAXABS: z = z**2 + c else: return k # diverged after k trials return MAXIT # did not diverge, for x in range(X): for y in range(Y): re = rscale * x - rlim # complex number represented im = iscale * y - ilim # by the (x,y) coordinate m[x][y] = numit(re,im) # get the color for (x,y) imshow(m) # Colored plot using the two dimensional matrix show() pycode-browser-1.02/Code/Maths/kin.py000077500000000000000000000002421254404601000174560ustar00rootroot00000000000000#Example: kin.py x = input('Enter an integer ') y = input('Enter one more ') print 'The sum is ', x + y s = raw_input('Enter a String ') print 'You entered ', s pycode-browser-1.02/Code/Maths/kin2.py000077500000000000000000000003051254404601000175400ustar00rootroot00000000000000#Example: kin2.py x,y = input('Enter x and y separated by comma ') print 'The sum is ', x + y s = raw_input('Enter a decimal number ') print 'Your input string x 2 = ', s a = float(s) print a * 2 pycode-browser-1.02/Code/Maths/lissa.py000077500000000000000000000002611254404601000200110ustar00rootroot00000000000000#Example lissa.py from pylab import * a = 2 b = 3 t= linspace(0, 2*pi,100) x = a * sin(2*t) y = b * cos(t) plot(x,y) x = a * sin(3*t) y = b * cos(2*t) plot(x,y) show() pycode-browser-1.02/Code/Maths/mathlocal.py000077500000000000000000000001321254404601000206370ustar00rootroot00000000000000#Example mathlocal.py from math import sin # sin is imported as local print sin(0.5) pycode-browser-1.02/Code/Maths/mathlocal2.py000077500000000000000000000001331254404601000207220ustar00rootroot00000000000000#Example mathlocal2.py from math import * # import everything from math print sin(0.5) pycode-browser-1.02/Code/Maths/mathsin.py000077500000000000000000000001161254404601000203400ustar00rootroot00000000000000#Example math.py import math print math.sin(0.5) # module_name.method_name pycode-browser-1.02/Code/Maths/mathsin2.py000077500000000000000000000001541254404601000204240ustar00rootroot00000000000000#Example math2.py import math as m # Give another madule name print m.sin(0.5) # Refer by the new name pycode-browser-1.02/Code/Maths/max.py000077500000000000000000000002421254404601000174620ustar00rootroot00000000000000#Example: max.py max = 0 while 1: # Infinite loop x = input('Enter a number ') if x > max: max = x if x == 0: print max break pycode-browser-1.02/Code/Maths/npsin.py000077500000000000000000000002001254404601000200160ustar00rootroot00000000000000#Example npsin.py from pylab import * x = linspace(-pi, pi , 200) y = sin(x) y1 = sin(x*x) plot(x,y) plot(x,y1,'r') show() pycode-browser-1.02/Code/Maths/numpy1.py000077500000000000000000000001461254404601000201310ustar00rootroot00000000000000#Example numpy1.py from numpy import * x = array( [1,2] ) # Make array from list print x , type(x) pycode-browser-1.02/Code/Maths/numpy2.py000077500000000000000000000003131254404601000201260ustar00rootroot00000000000000#Example numpy2.py from numpy import * a = arange(1.0, 2.0, 0.1) # start, stop & step print a b = linspace(1,2,11) print b c = ones(5) print c d = zeros(5) print d e = random.rand(5) print e pycode-browser-1.02/Code/Maths/numpy3.py000077500000000000000000000002071254404601000201310ustar00rootroot00000000000000#Example numpy3.py from numpy import * a = [ [1,2] , [3,4] ] # make a list of lists x = array(a) # and convert to an array print a pycode-browser-1.02/Code/Maths/numpy4.py000077500000000000000000000001611254404601000201310ustar00rootroot00000000000000#Example numpy4.py from numpy import * a = arange(10) print a a = a.reshape(5,2) # 5 rows and 2 columns print a pycode-browser-1.02/Code/Maths/numpy5.py000077500000000000000000000002201254404601000201260ustar00rootroot00000000000000#Example numpy5.py from numpy import * a = array([ [1.0, 2.0] , [3.0, 4.0] ]) # make a numpy array print a print a * 5 print a * a print a / a pycode-browser-1.02/Code/Maths/oper.py000077500000000000000000000001621254404601000176430ustar00rootroot00000000000000#Example oper.py from numpy import * a = array([[2,3], [4,5]]) b = array([[1,2], [3,0]]) print a + b print a * b pycode-browser-1.02/Code/Maths/pants.py000077500000000000000000000012261254404601000200250ustar00rootroot00000000000000class clothing: def __init__(self,colour, rate): self.colour = colour self.rate = rate def getcolour(self): return self.colour def __str__(self): return '%s Cloth at Rs. %6.2f per sqmtr'%(self.colour, self.rate) class pants(clothing): def __init__(self, cloth, size, labour): self.labour = labour self.size = size clothing.__init__(self,cloth.colour, cloth.rate) def getcost(self): return self.rate * self.size + self.labour costlyred = clothing('red', 100.0) smallpant = pants(costlyred, 1.5, 100) print smallpant.getcost() print smallpantpycode-browser-1.02/Code/Maths/pickledump.py000077500000000000000000000001611254404601000210320ustar00rootroot00000000000000#Example pickle.py import pickle f = open('test.pck' , 'w') pickle.dump(12.3, f) # write a float type f.close() pycode-browser-1.02/Code/Maths/pickleload.py000077500000000000000000000002221254404601000210020ustar00rootroot00000000000000#Example pickle2.py import pickle f = open('test.pck' , 'r') x = pickle.load(f) print x , type(x) # check the type of data read f.close() pycode-browser-1.02/Code/Maths/plot1.py000077500000000000000000000001101254404601000177260ustar00rootroot00000000000000#Example plot1.py from pylab import * data = [1,2,5] plot(data) show() pycode-browser-1.02/Code/Maths/plot2.py000077500000000000000000000001201254404601000177300ustar00rootroot00000000000000#Example plot2.py from pylab import * x = [1,2,5] y = [4,5,6] plot(x,y) show() pycode-browser-1.02/Code/Maths/plot3.py000077500000000000000000000002071254404601000177370ustar00rootroot00000000000000#Example plot3.py from pylab import * x = [1,2,5] y = [4,5,6] plot(x,y,'ro') xlabel('x-axis') ylabel('y-axis') axis([0,6,1,7]) show() pycode-browser-1.02/Code/Maths/plot4.py000077500000000000000000000002061254404601000177370ustar00rootroot00000000000000#Example plot4.py from pylab import * t = arange(0.0, 5.0, 0.2) plot(t, t**2,'x') # t^{2} plot(t, t**3,'ro') # t^{3} show() pycode-browser-1.02/Code/Maths/polar.py000077500000000000000000000002131254404601000200100ustar00rootroot00000000000000#Example polar.py from pylab import * th = linspace(0,2*pi,100) r = 5 * ones(100) # radius = 5 axis([0, 2*pi, 0, 10]) polar(th,r) show() pycode-browser-1.02/Code/Maths/poly.py000077500000000000000000000002601254404601000176600ustar00rootroot00000000000000#Example poly.py from pylab import * a = poly1d([3,4,5]) b = poly1d([6,7]) c = a * b + 5 d = c/a print a print b print a * b print d[0], d[1] print a.deriv() print a.integ() pycode-browser-1.02/Code/Maths/polyplot.py000077500000000000000000000002201254404601000205530ustar00rootroot00000000000000#Example polyplot.py from pylab import * x = linspace(-pi, pi, 100) a = poly1d([-1.0/5040,0,1.0/120,0,-1.0/6,0,1,0]) y = a(x) plot(x,y) show() pycode-browser-1.02/Code/Maths/power.py000077500000000000000000000001331254404601000200300ustar00rootroot00000000000000def power(mant, exp = 2.0): return mant ** exp print power(5., 3) print power(4.) pycode-browser-1.02/Code/Maths/reshape.py000077500000000000000000000001261254404601000203250ustar00rootroot00000000000000#Example reshape.py from numpy import * a = arange(20) b = reshape(a, [4,5]) print b pycode-browser-1.02/Code/Maths/rfile.py000077500000000000000000000001071254404601000177760ustar00rootroot00000000000000#Example rfile.py f = open('test.dat' , 'r') print f.read() f.close() pycode-browser-1.02/Code/Maths/rfile2.py000077500000000000000000000002311254404601000200560ustar00rootroot00000000000000#Example rfile2.py f = open('test.dat' , 'r') print f.read(7) # get first seven characters print f.read() # get the remaining ones f.close() pycode-browser-1.02/Code/Maths/rfile3.py000077500000000000000000000004041254404601000200610ustar00rootroot00000000000000#Example rfile3.py f = open('data.dat' , 'r') while 1: # infinite loop s = f.readline() if s == '' : # Empty string means end of file break # terminate the loop m = int(s) # Convert to integer print m * 5 f.close() pycode-browser-1.02/Code/Maths/rk4.py000077500000000000000000000010401254404601000173720ustar00rootroot00000000000000''' Solving initial value problem using 4th order Runge-Kutta method. Sine function is calculated using its derivative, cosine. ''' import math def rk4(x, y, yprime, dx = 0.01): # x, y , derivative, stepsize k1 = dx * yprime(x) k2 = dx * yprime(x + dx/2.0) k3 = dx * yprime(x + dx/2.0) k4 = dx * yprime(x + dx) return y + ( k1/6 + k2/3 + k3/3 + k4/6 ) h = 0.01 # stepsize x = 0.0 # initail values y = 0.0 while x < math.pi: print x, y, math.sin(x) y = rk4(x,y,math.cos) # Runge-Kutta method x = x + h pycode-browser-1.02/Code/Maths/second.py000077500000000000000000000003101254404601000201440ustar00rootroot00000000000000#Example: second.py s = 'My name' print s[0] # will print M print s[3] print s[-1] # will print the last element a = [12, 3.4, 'haha'] # List type print a, type(a) print a[0] pycode-browser-1.02/Code/Maths/series_sin.py000077500000000000000000000004631254404601000210450ustar00rootroot00000000000000#Example series_sin.py from pylab import * from scipy import factorial x = linspace(-pi, pi, 50) y = zeros(50) for n in range(5): term = (-1)**(n) * (x**(2*n+1)) / factorial(2*n+1) y = y + term plot(x,term) plot(x, y, '+b') plot(x, sin(x),'xr') # compare with the real one show() pycode-browser-1.02/Code/Maths/sine.py000077500000000000000000000001461254404601000176360ustar00rootroot00000000000000#Example sine.py import math x = 0.0 while x < math.pi * 4: print x , math.sin(x) x = x + 0.1pycode-browser-1.02/Code/Maths/table.py000077500000000000000000000001071254404601000177640ustar00rootroot00000000000000#Example: table.py x = 1 while x <= 10: print x * 8 x = x + 1 pycode-browser-1.02/Code/Maths/test.pck000077500000000000000000000000001254404601000177710ustar00rootroot00000000000000pycode-browser-1.02/Code/Maths/third.py000077500000000000000000000003551254404601000200140ustar00rootroot00000000000000#Example: third.py s = [3, 3.5, 234] # make a list s[2] = 'haha' # Change an element a = (4, 3.2, 'hi') # Tuple type a[0] = 6 # Will give ERROR x = 'myname' # String type x[1] = 2 # Will give ERROR pycode-browser-1.02/Code/Maths/trapez.py000077500000000000000000000004471254404601000202110ustar00rootroot00000000000000from math import * def sqr(a): return a**2 def trapez(f, a, b, n): h = (b-a) / n sum = f(a) for i in range (1,n): sum = sum + 2 * f(a + h * i) sum = sum + f(b) return 0.5 * h * sum print trapez(sin,0.,pi,100) print trapez(sqr,0.,2.,100) print trapez(sqr,0,2,100) # Why the error ? pycode-browser-1.02/Code/Maths/vdiff.py000077500000000000000000000004071254404601000177760ustar00rootroot00000000000000#Example vdiff.py from pylab import * def f(x): return sin(x) def deriv(x,dx=0.005): df = f(x+dx/2)-f(x-dx/2) return df/dx vecderiv = vectorize(deriv) x = linspace(-2*pi, 2*pi, 200) y = vecderiv(x) plot(x,y,'+') plot(x,cos(x)) show() pycode-browser-1.02/Code/Maths/wfile.py000077500000000000000000000001301254404601000177770ustar00rootroot00000000000000#Example wfile.py f = open('test.dat' , 'w') f.write ('This is a test file') f.close() pycode-browser-1.02/Code/Maths/wfile2.py000077500000000000000000000001671254404601000200730ustar00rootroot00000000000000#Example wfile2.py f = open('data.dat' , 'w') for k in range(1,4): s = '%3d\n' %(k) f.write(s) f.close() pycode-browser-1.02/Code/Physics/000077500000000000000000000000001254404601000166705ustar00rootroot00000000000000pycode-browser-1.02/Code/Physics/Mechanics/000077500000000000000000000000001254404601000205625ustar00rootroot00000000000000pycode-browser-1.02/Code/Physics/Mechanics/cf3d.py000077500000000000000000000011101254404601000217470ustar00rootroot00000000000000import math from visual import * base = box (pos=(0,-5,0), length=15, height=0.01, width=0.01, color=color.black) m1 = sphere (pos=(0,0,0), radius=1.0, color=color.blue) m2 = sphere (pos=(4,0,0), radius=0.3, color=color.red) G = -1.0 M = 1000.0 m = 1.0 dt = 0.01 x = math.sqrt(12.5) y = x vx = 10.0 vy = -vx+2 a = [] b = [] while 1: rate(50) r = sqrt(x**2 + y**2) v = sqrt(vx**2 + vy**2) F = G*M*m/r**2 + 0.7 * v Fx = F * x/r Fy = F * y/r vx = vx + Fx/m * dt vy = vy + Fy/m * dt x = x + vx * dt y = y + vy * dt m2.pos=(x,y,0) a.append(x) b.append(y) pycode-browser-1.02/Code/Physics/Mechanics/spring1.py000077500000000000000000000011511254404601000225200ustar00rootroot00000000000000#spring1.py from pylab import * t = 0.0 # Stating time dt = 0.01 # value of time increment x = 10.0 # initial position v = 0.0 # initial velocity k = 10.0 # spring constant m = 2.0 # mass of the oscillating object tm = [] # Empty lists to store time, velocity and displacement vel = [] dis = [] while t < 5: f = -k * x # Try (-k*x - 0.5 * v) to add damping v = v + (f/m ) * dt # dv = a.dt x = x + v * dt # dS = v.dt t = t + dt tm.append(t) vel.append(v) dis.append(x) plot(tm,vel) plot(tm,dis) show() pycode-browser-1.02/Code/Physics/Mechanics/spring2.py000077500000000000000000000033701254404601000225260ustar00rootroot00000000000000""" spring2.py The rk4_two() routine in this program does a two step integration using an array method. The current x and xprime values are kept in a global list named 'val'. val[0] = current position; val[1] = current velocity The results are compared with analytically calculated values. """ from pylab import * def accn(t, val): force = -spring_const * val[0] - damping * val[1] return force/mass def vel(t, val): return val[1] def rk4_two(t, h): # Time and Step value global xxp # x and xprime values in a 'xxp' k1 = [0,0] # initialize 5 empty lists. k2 = [0,0] k3 = [0,0] k4 = [0,0] tmp= [0,0] k1[0] = vel(t,xxp) k1[1] = accn(t,xxp) for i in range(2): # value of functions at t + h/2 tmp[i] = xxp[i] + k1[i] * h/2 k2[0] = vel(t + h/2, tmp) k2[1] = accn(t + h/2, tmp) for i in range(2): # value of functions at t + h/2 tmp[i] = xxp[i] + k2[i] * h/2 k3[0] = vel(t + h/2, tmp) k3[1] = accn(t + h/2, tmp) for i in range(2): # value of functions at t + h tmp[i] = xxp[i] + k3[i] * h k4[0] = vel(t+h, tmp) k4[1] = accn(t+h, tmp) for i in range(2): # value of functions at t + h xxp[i] = xxp[i] + ( k1[i] + \ 2.0*k2[i] + 2.0*k3[i] + k4[i]) * h/ 6.0 t = 0.0 # Stating time h = 0.01 # Runge-Kutta step size, time increment xxp = [2.0, 0.0] # initial position & velocity spring_const = 100.0 # spring constant mass = 2.0 # mass of the oscillating object damping = 0.0 tm = [0.0] # Lists to store time, position & velocity x = [xxp[0]] xp = [xxp[1]] xth = [xxp[0]] while t < 5: rk4_two(t,h) # Do one step RK integration t = t + h tm.append(t) xp.append(xxp[1]) x.append(xxp[0]) th = 2.0 * cos(sqrt(spring_const/mass)* (t)) xth.append(th) plot(tm,x) plot(tm,xth,'+') show() pycode-browser-1.02/Code/Physics/Mechanics/spring3d.py000077500000000000000000000016061254404601000226730ustar00rootroot00000000000000#spring3d.py from visual import * base = box (pos=(0,-1,0), length=16, height=0.1, width=4, color=color.blue) wall = box (pos=(0,0,0), length=0.1, height=2, width=4, color=color.white) ball = sphere (pos=(4,0,0), radius=1, color=color.red) spring = helix(pos=(0,0,0), axis=(4,0,0), radius=0.5, color=color.red) ball2 = sphere (pos=(-4,0,0), radius=1, color=color.green) spring2 = helix(pos=(0,0,0), axis=(-4,0,0), radius=0.5, color=color.green) t = 0.0 dt = 0.01 x1 = 2.0 x2 = -2.0 v1 = 0.0 v2 = 0.0 k = 1000.0 m = 1.0 while 1: rate(20) f1 = -k * x1 v1 = v1 + (f1/m ) * dt # Acceleration = Force / mass ; dv = a.dt f2 = -k * x2 - v2 # damping proportional to velocity v2 = v2 + (f2/m ) * dt # Acceleration = Force / mass ; dv = a.dt x1 = x1 + v1 * dt x2 = x2 + v2 * dt t = t + dt spring.length = 4 + x1 ball.x = x1 + 4 spring2.length = 4 - x2 ball2.x = x2 - 4 pycode-browser-1.02/Code/Physics/bes1.py000077500000000000000000000003151254404601000200760ustar00rootroot00000000000000from pylab import * from scipy import * def vj0(xarray): y = [] for x in xarray: val = special.j0(x) # Compute Jo y.append(val) return y a = linspace(0,10,100) b = vj0(a) plot(a,b) show() pycode-browser-1.02/Code/Physics/bes2.py000077500000000000000000000003531254404601000201010ustar00rootroot00000000000000from pylab import * from scipy import * def vjn(n,xarray): y = [] for x in xarray: val = special.jn(n,x) # Compute Jn(x) y.append(val) return y a = linspace(0,10,100) for n in range(5): b = vjn(n,a) plot(a,b) show() pycode-browser-1.02/Code/Physics/bes3.py000077500000000000000000000011301254404601000200740ustar00rootroot00000000000000from scipy import * from scipy import special import pylab def jn(n,x): jn = 0.0 for k in range(30): num = (-1)**k * (x/2)**(2*k) den = factorial(k)*factorial(n+k) jn = jn + num/den return jn * (x/2)**n def vjn_local(n,xarray): y = [] for x in xarray: val = jn(n,x) # Jn(x) using our function y.append(val) return y def vjn(n,xarray): y = [] for x in xarray: val = special.jn(n,x) # Compute Jn(x) y.append(val) return y a = linspace(0,10,100) for n in range(2): b = vjn(n,a) c = vjn_local(n,a) pylab.plot(a,b) pylab.plot(a,c,marker = '+') pylab.show() pycode-browser-1.02/Code/Physics/cf3d.py000077500000000000000000000011101254404601000200550ustar00rootroot00000000000000import math from visual import * base = box (pos=(0,-5,0), length=15, height=0.01, width=0.01, color=color.black) m1 = sphere (pos=(0,0,0), radius=1.0, color=color.blue) m2 = sphere (pos=(4,0,0), radius=0.3, color=color.red) G = -1.0 M = 1000.0 m = 1.0 dt = 0.01 x = math.sqrt(12.5) y = x vx = 10.0 vy = -vx+2 a = [] b = [] while 1: rate(50) r = sqrt(x**2 + y**2) v = sqrt(vx**2 + vy**2) F = G*M*m/r**2 + 0.7 * v Fx = F * x/r Fy = F * y/r vx = vx + Fx/m * dt vy = vy + Fy/m * dt x = x + vx * dt y = y + vy * dt m2.pos=(x,y,0) a.append(x) b.append(y) pycode-browser-1.02/Code/Physics/cp_em.py000077500000000000000000000017051254404601000203330ustar00rootroot00000000000000#cp_em.py -- Motion of charged particle in E & M fields from numpy import * import pylab as p import matplotlib.axes3d as p3 Ex = 0.0 # Components of Applied Electric Field Ey = 0.0 Ez = 2.0 Bx = 0.0 # Magnetic field By = 0.0 Bz = 2.0 m = 2.0 # Mass of the particle q = 5.0 # Charge x = 0.0 # Components of initial position and velocity y = 0.0 z = 0.0 vx = 20.0 vy = 0.0 vz = 0.0 a = [] b = [] c = [] t = 0.0 dt = 0.01 while t < 6: # trace until time reaches 6 units Fx = q * (Ex + (vy * Bz) - (vz * By) ) vx = vx + Fx/m * dt # Acceleration = F/m; dv = a.dt Fy = q * (Ey - (vx * Bz) + (vz * Bx) ) vy = vy + Fy/m * dt Fz = q * (Ez + (vx * By) - (vy * Bx) ) vz = vz + Fz/m * dt # print vx,vy, sqrt(vx**2+vy**2) x = x + vx * dt y = y + vy * dt z = z + vz * dt a.append(x) b.append(y) c.append(z) t = t + dt fig=p.figure() ax = p3.Axes3D(fig) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_zlabel('Z') ax.plot3D(a,b,c) p.show() pycode-browser-1.02/Code/Physics/diff1.py000077500000000000000000000007421254404601000202410ustar00rootroot00000000000000from pylab import * dx = 0.1 # value of x increment x = arange(0,10, dx) y = sin(x) yprime = [] # empty list for k in range(99): # from 100 points we get 99 diference values dy = y[k+1]-y[k] yprime.append(dy/dx) x1 = x[:-1] # A new array without the last element x1 = x1 + dx/2 # The derivative corresponds to the middle point plot(x1, yprime, '+') plot(x1, cos(x1)) # Cross check with the analitic value show() pycode-browser-1.02/Code/Physics/fit1.py000077500000000000000000000007371254404601000201170ustar00rootroot00000000000000from pylab import * from scipy import * from scipy.optimize import leastsq def err_func(p,y,x): A,k,theta = p return y - A*sin(2*pi*k*x+theta) def evaluate(x,p): return p[0] * sin(2*pi*p[1]*x+p[2]) ax = arange(0, 0.2, 0.001) A,k,theta = 5, 50.0, pi/3 y_true = A*sin(2*pi*k*ax+theta) y_meas = y_true + 0.2*randn(len(ax)) p0 = [6, 50.0, pi/3] plsq = leastsq(err_func,p0,args=(y_meas,ax)) print plsq plot(ax,y_true) plot(ax,y_meas,'o') plot(ax,evaluate(ax,plsq[0])) show() pycode-browser-1.02/Code/Physics/fit2.py000077500000000000000000000007371254404601000201200ustar00rootroot00000000000000from pylab import * from scipy import * from scipy.optimize import leastsq def err_func(p,y,x): A,k,theta = p return y - A*sin(2*pi*k*x+theta) def evaluate(x,p): return p[0] * sin(2*pi*p[1]*x+p[2]) ax = arange(0, 0.2, 0.001) A,k,theta = 5, 50.0, pi/3 y_true = A*sin(2*pi*k*ax+theta) y_meas = y_true + 0.2*randn(len(ax)) p0 = [6, 50.0, pi/3] plsq = leastsq(err_func,p0,args=(y_meas,ax)) print plsq plot(ax,y_true) plot(ax,y_meas,'o') plot(ax,evaluate(ax,plsq[0])) show() pycode-browser-1.02/Code/Physics/fourier1.py000077500000000000000000000002631254404601000210020ustar00rootroot00000000000000from pylab import * x = linspace(0.0, 2 * pi, 100) y = sin(x) for h in range(3,10,2): # Add even harmonics to get square wave y = y + sin(h*x)/h plot(x,y) show() pycode-browser-1.02/Code/Physics/hydro.py000077500000000000000000000012051254404601000203700ustar00rootroot00000000000000from numpy import * from scipy import special import pylab as p import matplotlib.axes3d as p3 phi = linspace(0, 2*pi, 50) theta = linspace(-pi/2, pi/2, 200) ax = [] ay = [] az = [] R = 1.0 for t in theta: polar = float(t) for k in phi: azim = float(k) sph = special.sph_harm(0,5,azim, polar) # Y(m,l,phi,theta) modulation = 0.2 * abs(sph) r = R * ( 1 + modulation) x = r*cos(polar)*cos(azim) y = r*cos(polar)*sin(azim) z = r*sin(polar) # print z # print x,y,z ax.append(x) ay.append(y) az.append(z) fig=p.figure() f = p3.Axes3D(fig) f.set_xlabel('X') f.set_ylabel('Y') f.set_zlabel('Z') f.scatter3D(ax,ay,az) p.show() pycode-browser-1.02/Code/Physics/legen1.py000077500000000000000000000002441254404601000204200ustar00rootroot00000000000000from scipy import linspace, special import pylab x = linspace(-1,1,100) for n in range(1,6): leg = special.legendre(n) y = leg(x) pylab.plot(x,y) pylab.show() pycode-browser-1.02/Code/Physics/numpy1.py000077500000000000000000000003361254404601000205000ustar00rootroot00000000000000from numpy import * a = array( [ 10, 20, 30, 40 ] ) # create an array out of a list print a print a * 3 print a / 3 b = array([1.0, 2.0, 3.0, 4.0]) c = a + b a = array([1,2,3,4], dtype='float') a = array([[1,2],[3,4]]) pycode-browser-1.02/Code/Physics/numpy2.py000077500000000000000000000003261254404601000205000ustar00rootroot00000000000000from pylab import * from numpy import * x = linspace(0.0, 2 * pi, 100) s = sin(x) # sine of each element of 'x' is taken c = cos(x) # cosine of each element of 'x' is taken plot(x,s) plot(x,c) show() pycode-browser-1.02/Code/Physics/numpy3.py000077500000000000000000000002511254404601000204760ustar00rootroot00000000000000from pylab import * x = linspace(0.0, 2 * pi, 100) plot(sin(x), cos(x), 'b') # plot lissagous figures plot(sin(2*x), cos(x),'r') plot(sin(x), cos(2*x),'y') show() pycode-browser-1.02/Code/Physics/plot1.py000077500000000000000000000000651254404601000203050ustar00rootroot00000000000000from pylab import * x = range(10) plot(x) show() pycode-browser-1.02/Code/Physics/plot2.py000077500000000000000000000001721254404601000203050ustar00rootroot00000000000000from pylab import * x = range(10,20) y = range(40,50) plot(x,y, marker = '+', color = 'red', markersize = 10) show() pycode-browser-1.02/Code/Physics/plot3.py000077500000000000000000000003021254404601000203010ustar00rootroot00000000000000from pylab import * x = [] # empty lists y = [] for k in range(100): ang = 0.1 * k sang = sin(ang) x.append(ang) y.append(sang) plot(x,y) show() pycode-browser-1.02/Code/Physics/plot3d1.py000077500000000000000000000004211254404601000205300ustar00rootroot00000000000000from numpy import * import pylab as p import matplotlib.axes3d as p3 w = v = u = linspace(0, 4*pi, 100) fig=p.figure() ax = p3.Axes3D(fig) ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_zlabel('Z') ax.plot3D(u,v,sin(w)) ax.plot3D(u,sin(v),w) ax.plot3D(sin(u),v,u) p.show() pycode-browser-1.02/Code/Physics/poly1.py000077500000000000000000000004431254404601000203120ustar00rootroot00000000000000import scipy import pylab a = scipy.poly1d([3,4,5]) print a, ' is the polynomial' print a*a, 'is its square' print a.deriv(), ' is its derivative' print a.integ(), ' is its integral' print a(0.5), 'is its value at x = 0.5' x = scipy.linspace(0,5,100) b = a(x) pylab.plot(x,b) pylab.show() pycode-browser-1.02/Code/Physics/pyphy.lyx000077500000000000000000001763431254404601000206200ustar00rootroot00000000000000#LyX 1.4.2 created this file. For more info see http://www.lyx.org/ \lyxformat 245 \begin_document \begin_header \textclass article \language english \inputencoding auto \fontscheme default \graphics default \paperfontsize default \spacing other 1.2 \papersize default \use_geometry false \use_amsmath 1 \cite_engine basic \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \end_header \begin_body \begin_layout Title Python in Physics Education \end_layout \begin_layout Author ajith@iuac.res.in \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard Mathematics is considered as the language of physics. Physical laws are expressed as mathematical relations and they are solved under desired boundary conditions to obtain the results. To illustrate this point take the example of the relationship between electric and magnetic fields as represented by the Maxwell's equations \end_layout \begin_layout Standard \begin_inset Formula \[ \nabla\times E=-\frac{\partial B}{\partial t}\,;\,\,\nabla\times B=\mu_{0}J+\mu_{0}\varepsilon_{0}\frac{\partial E}{\partial t}\,;\,\nabla.E=\frac{\rho}{\varepsilon_{0}}\,;\nabla.B=0\] \end_inset \end_layout \begin_layout Standard In order to find out the field distribution inside a resonator cavity, we need to solve these equations by applying suitable boundary conditions dictated by the geometry of the cavity and electromagnetic properties of the material used for making it. For simple geometries, analytic solutions are possible but in most cases numerical methods are essential and that is one reason why computers are important for physics. \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none With the wide spread availability of personal computers, programming has been included in almost all Physics courses. Unfortunately the learning has been in separate compartments without any attempt to apply the acquired programming skills to understand physics in a better way. Partially this has been due to, in my view, the complexities of the chosen languages like FORTRAN, C, C++ etc. Another drawback was the lack of libraries to generate visual output. This writeup is an attempt to connect computer programming with physics learning. The language chosen is Python, due to its simplicity and the huge collection of libraries, mainly the ones for doing math and graphics. \family default \series default \shape default \size default \emph default \bar default \noun default \begin_inset Foot status open \begin_layout Standard All the Python programs listed in this article are on the Phoenix Live CD. To run them boot from the liveCD, in graphics mode open a terminal, change to directory 'phy' and type \end_layout \begin_layout Standard # python program_name.py \end_layout \end_inset \end_layout \begin_layout Section Python Basics \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none The first reaction from most of the readers would be like \begin_inset Quotes eld \end_inset Oh. No. One more programming language. I already had tough time learning the one know now \begin_inset Quotes erd \end_inset . Cool down, Python is not harmful as the name sounds. A high school student with average intelligence can pick it up within two weeks. What is there in a programming language anyway. It allows you to create variables, like elementary algebra, and allows you to manipulate them using arithmetic and logical operators. A sequence of such statements make the program. But most of the program are are not meant for running from the first line to the last line. Depending on the situation some lines may execute more than once and some may be totally skipped. This is done by using control flow statements for looping and decision making (while/for loops and if-elif-else kind of stuff). The variables belong to certain types like 'integer', 'float', 'string' etc. If you are new to programming refer to the book 'Snake wrangling for Kids', included in Phoenix CD. After that, as a test, read the code fragments below and if you can guess their outputs, you know enough to proceed with the material in this writeup. \end_layout \begin_layout Standard Example 1: test1.py \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a = 1 \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none b = 1.2 \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none c = 2 + 3j \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none d = 'i am a string' \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none e = [a, b, c, 'hello'] \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print d, b * c, e #what will be the output \end_layout \begin_layout LyX-Code \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Example 2: test2.py \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x = 1 \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none while x <= 10: \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print x * 8 \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x = x + 1 \end_layout \begin_layout LyX-Code \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Example 3: test3.py \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none s = 'hello world' \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none m = [1, 2.3, 33, 'ha'] \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none for k in s: \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print k \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none for k in m: \end_layout \begin_layout LyX-Code print m \end_layout \begin_layout LyX-Code \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Example 4: test4.py \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x = range(10) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print x \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none for i in x: \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none if i > 8: \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print i \end_layout \begin_layout LyX-Code \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Example 5: test5.py \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a = [12, 23] \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a.append(45) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print a \end_layout \begin_layout LyX-Code \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Example 6: test6.py ( Three different ways to import a module) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none import time \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print time.ctime() \end_layout \begin_layout LyX-Code from time import * \end_layout \begin_layout LyX-Code print time() \end_layout \begin_layout LyX-Code import time as t \end_layout \begin_layout LyX-Code print t.time() \end_layout \begin_layout LyX-Code from math import sin #imports only one function \end_layout \begin_layout LyX-Code print sin(2.0) \end_layout \begin_layout Standard Some modules may have several sub-packages containing functions. For example the 'Special functions' sub-package from 'scipy' module can be imported as \end_layout \begin_layout LyX-Code from scipy import special \end_layout \begin_layout LyX-Code print special.j0(0.2) # print values of Bessel function \end_layout \begin_layout Section Plotting Graphs \end_layout \begin_layout Standard While studying physics, we come across various \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none mathematical functions. In order to understand a function, we need to have some idea about its derivatives, integrals, location of zeros, maxima and minima, etc. A graphical representation conveys them quickly. To do graph plotting without much coding, we need to use some library ( called modules in Python). We will use Matplotlib (refer to the matplotlib user's guide on Phoenix CD for details). Let us start with a simple example (plot1.py). \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none from pylab import * \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x = range(10) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none plot(x) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none show() \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none The 'Matplotlib' package contains much more than simple plotting, and are available in a module named \family default \series default \shape italic \size default \emph default \bar default \noun default pylab \family roman \series medium \shape up \size normal \emph off \bar no \noun off . The first line imports all the functions from 'pylab'. The \family default \series default \shape italic \size default \emph default \bar default \noun default range() \family roman \series medium \shape up \size normal \emph off \bar no \noun off function generates a 'list' of ten numbers, that is passed on to the plot routine. When only a single list is given, \family default \series default \shape italic \size default \emph default \bar default \noun default plot() \family roman \series medium \shape up \size normal \emph off \bar no \noun off takes it as the y-coordinates and x-coordinates are taken from 0 to N-1, where N is the number of y-coordinates given. The last statement \family default \series default \shape italic \size default \emph default \bar default \noun default show() \family roman \series medium \shape up \size normal \emph off \bar no \noun off tells matplotlib to display the created graphs. You can specify both x and y coordinates to the plot routine. It is also possible to specify several other properties like the marker shape and color. (plot2.py) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none from pylab import * \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x = range(10,20) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none y = range(40,50) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none plot(x,y, marker = '+', color = 'red') \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none show() \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Well, enough of the 2D plots using 'list' of integers. How about plotting some mathematical functions. Let us plot a sine curve. (plot3.py) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code x = [] \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none # empty lists \end_layout \begin_layout LyX-Code y = [] \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none for k in range(100): \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none ang = 0.1 * k \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none sang = sin(ang) # sine function from pylab \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x.append(ang) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none y.append(sang) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none plot(x,y) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none show() \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none First we made two empty lists x and y. Then entered a \family default \series default \shape italic \size default \emph default \bar default \noun default for loop \family roman \series medium \shape up \size normal \emph off \bar no \noun off where the integer k goes from 0 to 99. For each value of 'k' we took '0.1 * k' as the angle and added to the list 'x'. The sine of each angle is added to the list 'y'. Doing this kind of coding is fine if your objective is to learn computer programming but we are trying to use it as a tool for learning physics. \end_layout \begin_layout Subsection NumPy and Scipy \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none To get rid of loops etc. we need data types likes arrays and matrices. Fortunately Python has such libraries, for example Numpy. For learning more about Numpy, refer to the Numpy documentation on the Phoenix CD or www.scipy.org website. \begin_inset Foot status collapsed \begin_layout Standard Numpy documentation is not complete, however one can go through the online help about numpy at its subpackages. Start Python interpreter from a Terminal, import the package and ask for help as shown below. \end_layout \begin_layout LyX-Code #python \end_layout \begin_layout LyX-Code >>>import numpy \end_layout \begin_layout LyX-Code >>>help(numpy) \end_layout \begin_layout LyX-Code >>>help(numpy.core) \end_layout \begin_layout LyX-Code \end_layout \end_inset Try out the simple examples shown below. (numpy1.py) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none from numpy import * \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a = array( [ 10, 20, 30, 40 ] ) # create an array from a list \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print a \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print a * 3 \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print a / 3 \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none b = array([1.0, 2.0, 3.0, 4.0]) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none c = a + b \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a = array([1,2,3,4], dtype='float') \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a = array([[1,2],[3,4]]) \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Note that the operations performed on an array is carried out on each element, we need not do it explicitly. Numpy has several functions for the automatic creation of array objects. Try the following lines to see how they work. \end_layout \begin_layout LyX-Code a = ones(10) \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code a = zeros(10) \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a= arange(0,10) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print a \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a = arange(0, 1, 0.1) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print a \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a = arange(0,10,1,dtype = 'float') \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print a \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none a = linspace(0,10,100) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none print a \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Both 'arange' and 'linspace' functions generate an array. The first and second arguments are the first and last elements. For 'arange' , the third argument is the increment and for 'linspace' it is the number of elements in the array. In the case of arange() the last element may not match with the second argument given. We will try to make use of them in order to plot some mathematical functions. (numpy2.) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code x = linspace(0.0, 2 * pi, 100) \end_layout \begin_layout LyX-Code s = sin(x) # sine of each element of 'x' is taken \end_layout \begin_layout LyX-Code c = cos(x) # cosine of each element of 'x' is taken \end_layout \begin_layout LyX-Code plot(x,s) \end_layout \begin_layout LyX-Code plot(x,c) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard Note that we have not imported Numpy explicitly. Pylab requires 'numpy' and importing pylab brings numpy automatically. The 'linspace' function generates an array of 100 equi-spaced values ranging from \begin_inset Formula $0\, to\,2\pi$ \end_inset . Also note that the \begin_inset Formula $sin()$ \end_inset function ,from Numpy, is a vectorized version. It takes an array as an argument, computes the sine of each element and returns the result in an array. Such features reduce the complexity of the program and allows us to concentrate on physics rather than the computer program. Comparing this with the previous program illustrates the simplicity achieved. Try another example generating lissagous figures, (numpy3.py) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none from pylab import * \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x = linspace(0.0, 2 * pi, 100) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none plot(sin(x), cos(x), 'b') # plot lissagous figures \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none plot(sin(2*x), cos(x),'r') \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none plot(sin(x), cos(2*x),'y') \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none show() \end_layout \begin_layout Subsection Fourier Series \end_layout \begin_layout Standard The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions. Fourier series serve many useful purposes, as manipulation and conceptualizatio n of the modal coefficients are often easier than with the original function. Areas of application include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. The example below shows how to generate a Square wave using this technique. To make the output better, increase the number of terms by changing the second argument of the range function. (fourier1.py) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code x = linspace(0.0, 2 * pi, 100) \end_layout \begin_layout LyX-Code y = sin(x) \end_layout \begin_layout LyX-Code for h in range(3,10,2): \end_layout \begin_layout LyX-Code y = y + sin(h*x) / h \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section Power Series \end_layout \begin_layout Standard Trignometric functions like sine and cosine sounds very familiar to all of us, due to our familiarity with them since high school days. However most of us would find it difficult to obtain the numerical values of sin(2.0), say, without trigonometric tables or a calculator. We know that differentiating a sine function twice will give you the original function, with a sign reversal, which implies \end_layout \begin_layout Standard \begin_inset Formula \[ \frac{d^{2}y}{dx^{2}}+y=0\] \end_inset \end_layout \begin_layout Standard which has a series solution of the form \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} y=a_{0}\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n}}{(2n)!}+a_{1}\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n+1}}{(2n+1)!}\label{eq:Trig Series}\end{equation} \end_inset \end_layout \begin_layout Standard These are the Maclaurin series for sine and cosine functions. The following code plots several terms of the sine series and their sum. Try varying the number of terms to see the deviation. (series_sin.py) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from scipy import * \end_layout \begin_layout LyX-Code x = linspace(-pi,pi,40) \end_layout \begin_layout LyX-Code a = zeros(40) \end_layout \begin_layout LyX-Code plot(x,sin(x)) \end_layout \begin_layout LyX-Code for n in range(1,5): \end_layout \begin_layout LyX-Code sign = (-1)**(n+1) \end_layout \begin_layout LyX-Code term = x**(2*n-1) / factorial(2*n-1) \end_layout \begin_layout LyX-Code a = a + sign * term \end_layout \begin_layout LyX-Code print n,sign \end_layout \begin_layout LyX-Code plot(x,term) \end_layout \begin_layout LyX-Code plot(x,a,'+') \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard The program given below evaluates the cosine function using this series and compares it with the result of the cos() function from the math library (that also must be doing the same). We have written a vectored version of the cosine function that can accept an array argument. You can study the accuracy of the results obtained by varying the number of terms included. (series_cos.py) \end_layout \begin_layout LyX-Code import scipy, pylab \end_layout \begin_layout LyX-Code def mycos(x): \end_layout \begin_layout LyX-Code res = 0.0 \end_layout \begin_layout LyX-Code for n in range(18): \end_layout \begin_layout LyX-Code res = res + (-1)**n * (x ** (2*n)) / scipy.factorial(2*n) \end_layout \begin_layout LyX-Code return res \end_layout \begin_layout LyX-Code def vmycos(ax): \end_layout \begin_layout LyX-Code y = [] \end_layout \begin_layout LyX-Code for x in ax: \end_layout \begin_layout LyX-Code y.append(mycos(x)) \end_layout \begin_layout LyX-Code return y \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code x = scipy.linspace(0,4*scipy.pi,100) \end_layout \begin_layout LyX-Code y = vmycos(x) \end_layout \begin_layout LyX-Code pylab.plot(x,y) \end_layout \begin_layout LyX-Code pylab.plot(x,scipy.cos(x),'+') \end_layout \begin_layout LyX-Code pylab.show() \end_layout \begin_layout Section 3D Plots \end_layout \begin_layout Standard Matplotlib has a subpackage that supports 3D plots. Some simple examples are given below and we will come back to it later. (plot3d1.py) \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code import pylab as p \end_layout \begin_layout LyX-Code import matplotlib.axes3d as p3 \end_layout \begin_layout LyX-Code w = v = u = linspace(0, 4*pi, 100) \end_layout \begin_layout LyX-Code fig=p.figure() \end_layout \begin_layout LyX-Code ax = p3.Axes3D(fig) \end_layout \begin_layout LyX-Code ax.set_xlabel('X') \end_layout \begin_layout LyX-Code ax.set_ylabel('Y') \end_layout \begin_layout LyX-Code ax.set_zlabel('Z') \end_layout \begin_layout LyX-Code ax.plot3D(u,v,sin(w)) \end_layout \begin_layout LyX-Code ax.plot3D(u,sin(v),w) \end_layout \begin_layout LyX-Code ax.plot3D(sin(u),v,u) \end_layout \begin_layout LyX-Code p.show() \end_layout \begin_layout LyX-Code \end_layout \begin_layout Section Special Functions \end_layout \begin_layout Standard Solving certain differential equations give rise to special functions like Bessel. Legendre, Spherical harmonic etc. depending upon the symmetry properties of the coordinate systems used. Now we will start using another package called 'Scipy'. \begin_inset Foot status collapsed \begin_layout Standard Refer to the Phoenix CD or www.scipy.org for scipy documentation. \end_layout \end_inset Scipy has subpackages for a large number of mathematical operations inclding the computation of various special functions. \end_layout \begin_layout Subsection Bessel Functions \end_layout \begin_layout Standard Let us start with plotting the Bessel functions, restricted to integral values of n, given by the expression \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} J_{n}=\left(\frac{x}{2}\right)^{n}\sum_{k=0}^{\infty}\frac{\left(-\right)^{k}\left(\frac{x}{2}\right)^{2k}}{\left(k!\right)\left(n+k\right)!}\label{eq:Bessel}\end{equation} \end_inset \end_layout \begin_layout Standard using the following Python program. It uses the 'Scipy' module. (bes1.py) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from scipy import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def vj0(xarray): \end_layout \begin_layout LyX-Code y = [] \end_layout \begin_layout LyX-Code for x in xarray: \end_layout \begin_layout LyX-Code val = special.j0(x) # Compute Jo \end_layout \begin_layout LyX-Code y.append(val) \end_layout \begin_layout LyX-Code return y \end_layout \begin_layout LyX-Code a = linspace(0,10,100) \end_layout \begin_layout LyX-Code b = vj0(a) \end_layout \begin_layout LyX-Code plot(a,b) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard Note that function calculating Bessel function does not accept array arguments. So we had to write a vectorized version of the same. The next example generalizes the computation to higher orders as shown below. (bes2.py) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from scipy import * \end_layout \begin_layout LyX-Code def vjn(n,xarray): \end_layout \begin_layout LyX-Code y = [] \end_layout \begin_layout LyX-Code for x in xarray: \end_layout \begin_layout LyX-Code val = special.jn(n,x) # Compute Jn(x) \end_layout \begin_layout LyX-Code y.append(val) \end_layout \begin_layout LyX-Code return y \end_layout \begin_layout LyX-Code a = linspace(0,10,100) \end_layout \begin_layout LyX-Code for n in range(5): \end_layout \begin_layout LyX-Code b = vjn(n,a) \end_layout \begin_layout LyX-Code plot(a,b) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard In the two examples above, the order in which pylab and scipy are imported matters due to the following reason. The function linspace() is there in both the packages. They do the same job but the elements of the array returned by scipy.linspace() are 'float' type but they are 'numpy.float64' in the case of numpy.linspace(). These sort of conflicts can arise when you are importing multiple libraries. The best way to avoid them is to use the following syntax. \end_layout \begin_layout LyX-Code import scipy \end_layout \begin_layout LyX-Code import numpy \end_layout \begin_layout LyX-Code a = scipy.linspace(0,1,10) # from scipy \end_layout \begin_layout LyX-Code b = numpy.linspace(0,1,10) # from numpy \end_layout \begin_layout LyX-Code c = linspace(0,1,10) # from numpy, last import \end_layout \begin_layout LyX-Code \end_layout \begin_layout Standard Instead of using the library function from scipy(), we can calculate the Bessel functions using the expression \begin_inset LatexCommand \ref{eq:Bessel} \end_inset and compare the results. Try changing the number of terms to see the deviations. (bes3.py) \end_layout \begin_layout LyX-Code from scipy import * \end_layout \begin_layout LyX-Code from scipy import special \end_layout \begin_layout LyX-Code import pylab \end_layout \begin_layout LyX-Code def jn(n,x): \end_layout \begin_layout LyX-Code jn = 0.0 \end_layout \begin_layout LyX-Code for k in range(30): \end_layout \begin_layout LyX-Code num = (-1)**k * (x/2)**(2*k) \end_layout \begin_layout LyX-Code den = factorial(k)*factorial(n+k) \end_layout \begin_layout LyX-Code jn = jn + num/den \end_layout \begin_layout LyX-Code return jn * (x/2)**n \end_layout \begin_layout LyX-Code def vjn_local(n,xarray): \end_layout \begin_layout LyX-Code y = [] \end_layout \begin_layout LyX-Code for x in xarray: \end_layout \begin_layout LyX-Code val = jn(n,x) # Jn(x) using our function \end_layout \begin_layout LyX-Code y.append(val) \end_layout \begin_layout LyX-Code return y \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def vjn(n,xarray): \end_layout \begin_layout LyX-Code y = [] \end_layout \begin_layout LyX-Code for x in xarray: \end_layout \begin_layout LyX-Code val = special.jn(n,x) # Compute Jn(x) \end_layout \begin_layout LyX-Code y.append(val) \end_layout \begin_layout LyX-Code return y \end_layout \begin_layout LyX-Code a = linspace(0,10,100) \end_layout \begin_layout LyX-Code for n in range(2): \end_layout \begin_layout LyX-Code b = vjn(n,a) \end_layout \begin_layout LyX-Code c = vjn_local(n,a) \end_layout \begin_layout LyX-Code pylab.plot(a,b) \end_layout \begin_layout LyX-Code pylab.plot(a,c,marker = '+') \end_layout \begin_layout LyX-Code pylab.show() \end_layout \begin_layout Subsection Polynomials \end_layout \begin_layout Standard Before proceeding with other special functions, let us have a look at the one dimensional polynomial class 'poly1d' of Scipy. You can define a polynomial by supplying the coefficient as a list. For example , the statement p = poly1d([1,2,3]) constructs the polynomial x**2 + 2 x + 3. The following example describe the various operations you can do with this class. (ploy1.py) \end_layout \begin_layout LyX-Code import scipy \end_layout \begin_layout LyX-Code import pylab \end_layout \begin_layout LyX-Code a = scipy.poly1d([3,4,5]) \end_layout \begin_layout LyX-Code print a, ' is the polynomial' \end_layout \begin_layout LyX-Code print a*a, 'is its square' \end_layout \begin_layout LyX-Code print a.deriv(), ' is its derivative' \end_layout \begin_layout LyX-Code print a.integ(), ' is its integral' \end_layout \begin_layout LyX-Code print a(0.5), 'is its value at x = 0.5' \end_layout \begin_layout LyX-Code x = scipy.linspace(0,5,100) \end_layout \begin_layout LyX-Code b = a(x) \end_layout \begin_layout LyX-Code pylab.plot(a,b) \end_layout \begin_layout LyX-Code pylab.show() \end_layout \begin_layout Standard Note that a polynomial can take an array argument for evaluation to return the results in an array. \begin_inset Foot status collapsed \begin_layout Standard To know more type help(poly1d) at the python prompt after importing scipy; \end_layout \begin_layout LyX-Code >>>import scipy \end_layout \begin_layout LyX-Code >>>help(scipy.poly1d) \end_layout \end_inset \end_layout \begin_layout Subsection Legendre Functions \end_layout \begin_layout Standard The Legendre polynomials, sometimes called Legendre functions of the first kind, are solutions to the Legendre differential equation \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} (1-z^{2})\frac{d^{2}w}{dz^{2}}-2z\frac{dw}{dz}+n(n+1)w=0\label{eq:Legendre}\end{equation} \end_inset If n is an integer, they are polynomials. The Legendre polynomials P_n(x) are illustrated using the Python program given below. (legen1.py) \end_layout \begin_layout LyX-Code from scipy import linspace, special \end_layout \begin_layout LyX-Code import pylab \end_layout \begin_layout LyX-Code x = linspace(-1,1,100) \end_layout \begin_layout LyX-Code for n in range(1,6): \end_layout \begin_layout LyX-Code leg = special.legendre(n) \end_layout \begin_layout LyX-Code y = leg(x) \end_layout \begin_layout LyX-Code pylab.plot(x,y) \end_layout \begin_layout LyX-Code pylab.show() \end_layout \begin_layout Subsection Spherical Harmonics \end_layout \begin_layout Standard In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplace's equation represented in a system of spherical coordinates. Spherical harmonics are important in many theoretical and practical application s, particularly in the computation of atomic electron configurations, the representation of the gravitational field, magnetic field of planetary bodies, characterization of the cosmic microwave background radiation etc.. Laplace's equation in spherical polar coordinates can be written as \end_layout \begin_layout Standard \begin_inset Formula \[ \nabla^{2}\Phi=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial\Phi}{\partial r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Phi}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}\Phi}{\partial\phi^{2}}\] \end_inset \end_layout \begin_layout Standard Separation of variables in \begin_inset Formula $r,\theta and\phi$ \end_inset coordinates gives the angular part of the solution as \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} Y_{l}^{m}(\theta,\phi)=\sqrt{\frac{(2\ell+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos\theta)e^{im\phi}\label{eq:Spherical harmonics}\end{equation} \end_inset \end_layout \begin_layout Standard This is a called a spherical harmonic function of degree \begin_inset Formula $\ell$ \end_inset and order \begin_inset Formula $m$ \end_inset , \begin_inset Formula $P_{\ell}^{m}(\theta)$ \end_inset \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \family default \series default \shape default \size default \emph default \bar default \noun default an associated Legendre function. The following program calculates the spherical harmonics for \begin_inset Formula $\ell=10,m=0$ \end_inset . Change the values of \begin_inset Formula $\ell andm$ \end_inset to observe the changes. (sph1.py) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from scipy import special \end_layout \begin_layout LyX-Code a_th = [] # list to store polar angle theta from -90 to + 90 deg \end_layout \begin_layout LyX-Code a_sph = [] # list to store absolute values if sperical harminics \end_layout \begin_layout LyX-Code phi = 0.0 # Fix azimuth, phi at zero \end_layout \begin_layout LyX-Code theta = -pi/2 # start theta from -90 deg \end_layout \begin_layout LyX-Code while theta < pi/2: \end_layout \begin_layout LyX-Code h = special.sph_harm(0,10, phi, theta) # (m, l , phi, theta) \end_layout \begin_layout LyX-Code a_sph.append(abs(h)) \end_layout \begin_layout LyX-Code a_th.append(theta * 180/pi) \end_layout \begin_layout LyX-Code theta = theta + 0.02 \end_layout \begin_layout LyX-Code plot(a_th,a_sph) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard The spherical harmonics are easily visualized by counting the number of zero crossings they possess in both the latitudinal and longitudinal directions. For the latitudinal direction, the associated Legendre functions possess \begin_inset Formula $\ell-\left|m\right|$ \end_inset zeros, whereas for the longitudinal direction, the trigonomentric sin and cos functions possess \begin_inset Formula $2\left|m\right|$ \end_inset zeros. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Standard \align center \begin_inset Graphics filename pics/sph_from_wp.eps width 6cm keepAspectRatio \end_inset \end_layout \begin_layout Caption Schematic representation of \begin_inset Formula $Y_{\ell}^{m}$ \end_inset on the unit sphere. \begin_inset Formula $Y_{\ell}^{m}$ \end_inset is equal to 0 along m great circles passing through the poles, and along l-m circles of equal latitude. The function changes sign each time it crosses one of these lines. \begin_inset LatexCommand \label{fig:Schematic-representation-of Spherical harmonics} \end_inset \end_layout \end_inset \end_layout \begin_layout Standard When the spherical harmonic order m is zero, the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. When \begin_inset Formula $\ell=\left|m\right|$ \end_inset , there are no zero crossings in latitude, and the functions are referred to as sectoral. For the other cases, the functions checker the sphere, and they are referred to as tesseral. The variation of a function on the surface of a sphere is shown in figure \begin_inset LatexCommand \ref{fig:Schematic-representation-of Spherical harmonics} \end_inset . The Python program given below gives a 3D plot of a sphere. The radius is modulated using the value of \begin_inset Formula $Y_{\ell}^{m}$ \end_inset so that we can visualize it. (sph2.py) \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code from scipy import special \end_layout \begin_layout LyX-Code import pylab as p \end_layout \begin_layout LyX-Code import matplotlib.axes3d as p3 \end_layout \begin_layout LyX-Code phi = linspace(0, 2*pi, 50) \end_layout \begin_layout LyX-Code theta = linspace(-pi/2, pi/2, 200) \end_layout \begin_layout LyX-Code ax = [] \end_layout \begin_layout LyX-Code ay = [] \end_layout \begin_layout LyX-Code az = [] \end_layout \begin_layout LyX-Code R = 1.0 \end_layout \begin_layout LyX-Code for t in theta: \end_layout \begin_layout LyX-Code polar = float(t) \end_layout \begin_layout LyX-Code for k in phi: \end_layout \begin_layout LyX-Code azim = float(k) \end_layout \begin_layout LyX-Code sph = special.sph_harm(0,5,azim, polar) # Y(m,l,phi,theta) \end_layout \begin_layout LyX-Code modulation = 0.2 * abs(sph) \end_layout \begin_layout LyX-Code r = R * ( 1 + modulation) \end_layout \begin_layout LyX-Code x = r*cos(polar)*cos(azim) \end_layout \begin_layout LyX-Code y = r*cos(polar)*sin(azim) \end_layout \begin_layout LyX-Code z = r*sin(polar) \end_layout \begin_layout LyX-Code ax.append(x) \end_layout \begin_layout LyX-Code ay.append(y) \end_layout \begin_layout LyX-Code az.append(z) \end_layout \begin_layout LyX-Code fig=p.figure() \end_layout \begin_layout LyX-Code f = p3.Axes3D(fig) \end_layout \begin_layout LyX-Code f.set_xlabel('X') \end_layout \begin_layout LyX-Code f.set_ylabel('Y') \end_layout \begin_layout LyX-Code f.set_zlabel('Z') \end_layout \begin_layout LyX-Code f.scatter3D(ax,ay,az) \end_layout \begin_layout LyX-Code p.show() \end_layout \begin_layout LyX-Code \end_layout \begin_layout Section Numerical Differentiation \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none Simple Numerical Differentiation For a function y=f(x) , the derivative \begin_inset Formula $dy/dx$ \end_inset can be obtained numerically. We take the difference between consecutive elements \begin_inset Formula $dy$ \end_inset and divide it with the x increment \begin_inset Formula $dx$ \end_inset . The following program create an array of \begin_inset Formula $sines$ \end_inset and find the derivative numerically. The result is compared with the \begin_inset Formula $cosines$ \end_inset . (diff1.py) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none from pylab import * \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none dx = 0.1 # value of x increment \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x = arange(0,10, dx) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none y = sin(x) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none yprime = [] # empty list \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none for k in range(99): # from 100 points we get 99 diference values \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none dy = y[k+1]-y[k] \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none yprime.append(dy/dx) \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x1 = x[:-1] # A new array without the last element \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none x1 = x1 + dx/2 # The derivative corresponds to the middle point \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none plot(x1, yprime, '+') \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none plot(x1, cos(x1)) # Cross check with the analytic value \end_layout \begin_layout LyX-Code \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none show() \end_layout \begin_layout Standard (to be modified later) \end_layout \begin_layout Section Solving Differential Equations \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none If we know the value of a function \begin_inset Formula $y=f(x)$ \end_inset and its derivative at some particular value of 'x' , we can calculate the value of the function at a nearby point \begin_inset Formula $x+dx$ \end_inset . \begin_inset Formula $Y_{n+1}$ \end_inset = \begin_inset Formula $Y_{n}$ \end_inset + \begin_inset Formula $dx.(dy/dx)$ \end_inset . Integrating a function using this method is not very efficient compared to the more sophisticated methods like fourth order Runge-Kutta. However our objective is to just understand how numerical integration works. We will trace the 'sine' function, that can be \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{d(\sin(x))}{dx}=\cos(x)\label{eq:sine}\end{equation} \end_inset \end_layout \begin_layout Subsection Runge-Kutta Integration \end_layout \begin_layout Standard In the previous section we have used the Euler method that uses the formula \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} y_{n+1}=y_{n}+hf(x_{n},y_{n})\label{eq:Euler}\end{equation} \end_inset \end_layout \begin_layout Standard to evaluate the value of 'y' at point 'n+1' using the known value at 'n'. The formula is asymmetrical since it uses the derivative information at the beginning of the interval. In Fourth order Runge-Kutta method, for each step the derivative is evaluated four times; once at the initial point, twice at two trial midpoints, once at trial a endpoint. The final value is evaluated from these derivatives using the equations \end_layout \begin_layout Standard \begin_inset Formula \[ k_{1}=hf(x_{n},y_{n})\] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ k_{2}=hf(x_{n}+\frac{h}{2},y_{n}+\frac{k_{1}}{2})\] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ k_{3}=hf(x_{n}+\frac{h}{2},y_{n}+\frac{k_{2}}{2})\] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ k_{4}=hf(x_{n}+h,y_{n}+k_{3})\] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ y_{n+1}=y_{n}+\frac{k_{1}}{6}+\frac{k_{2}}{3}+\frac{k_{3}}{3}+\frac{k_{4}}{6}\] \end_inset \end_layout \begin_layout Standard In numerical analysis, the Runge-Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta. For more details on fouth order Runge-Kutta method refer to the file 'rk.pdf' on the Phoenix Live CD. The following program makes the sine curve using \shape italic fourth order RK method \shape default . At any point the derivative of the curve is cos(t) and we use this information to estimate the next point. (rk_sin.py) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def func(t): \end_layout \begin_layout LyX-Code return cos(t) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def rk4(s,t): \end_layout \begin_layout LyX-Code k1 = dt * func(t) \end_layout \begin_layout LyX-Code k2 = dt * func(t + dt/2.0) \end_layout \begin_layout LyX-Code k3 = dt * func(t + dt/2.0) \end_layout \begin_layout LyX-Code k4 = dt * func(t + dt) \end_layout \begin_layout LyX-Code return s + ( k1/6 + k2/3 + k3/3 + k4/6 ) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code t = 0.0 # Stating angle \end_layout \begin_layout LyX-Code dt = 0.01 # value of angle increment \end_layout \begin_layout LyX-Code val = 0.0 # value of the 'sine' function at t = 0 \end_layout \begin_layout LyX-Code tm = [0.0] # List to store theta values \end_layout \begin_layout LyX-Code res = [0.0] # RK4 results \end_layout \begin_layout LyX-Code while t < 10: \end_layout \begin_layout LyX-Code val = rk4(val,t) # get the new value \end_layout \begin_layout LyX-Code t = t + dt \end_layout \begin_layout LyX-Code tm.append(t) \end_layout \begin_layout LyX-Code res.append(val) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code plot(tm, res,'+') \end_layout \begin_layout LyX-Code plot(tm, sin(tm)) # compare with actual curve \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section Physics Simulations \end_layout \begin_layout Standard Using the numerical methods discussed in the previous sections we will try to solve some simple problems in physics, like calculating the trajectory of an object in a known force field. \end_layout \begin_layout Subsection Mass-Spring Problem \end_layout \begin_layout Standard As a simple example let us \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none take the "Mass-Spring" problem. A mass is attached to one end of a spring whose other end is fixed. If we displace the mass from the equilibrium position by \begin_inset Formula $x$ \end_inset , the force is given by the relation \begin_inset Formula $F=-kx$ \end_inset , where \begin_inset Formula $k$ \end_inset is the spring constant. If we release the mass at this point, it will start oscillating. The objective is to plot the displacement of the mass as a function of time. The force is known at the initial point \begin_inset Formula $-kx$ \end_inset , the acceleration of the mass when it is released is given by Newton's equation \begin_inset Formula $F=ma$ \end_inset , where m is the mass of the object. The initial position is Xo and initial velocity Vo is zero. We can integrate the acceleration to find out the velocity after a small time interval 'dt'. Similarly, integrating the velocity will give the displacement. (spring1.py) \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code t = 0.0 # Stating time \end_layout \begin_layout LyX-Code dt = 0.01 # value of time increment \end_layout \begin_layout LyX-Code x = 10.0 # initial position \end_layout \begin_layout LyX-Code v = 0.0 # initial velocity \end_layout \begin_layout LyX-Code k = 10.0 # spring constant \end_layout \begin_layout LyX-Code m = 2.0 # mass of the oscillating object \end_layout \begin_layout LyX-Code tm = [] # Empty lists to store time, velocity and displacement \end_layout \begin_layout LyX-Code vel = [] \end_layout \begin_layout LyX-Code dis = [] \end_layout \begin_layout LyX-Code while t < 5: \end_layout \begin_layout LyX-Code f = -k * x # Try (-k*x - 0.5 * v) to add damping \end_layout \begin_layout LyX-Code v = v + (f/m ) * dt # dv = a.dt \end_layout \begin_layout LyX-Code x = x + v * dt # dS = v.dt \end_layout \begin_layout LyX-Code t = t + dt \end_layout \begin_layout LyX-Code tm.append(t) \end_layout \begin_layout LyX-Code vel.append(v) \end_layout \begin_layout LyX-Code dis.append(x) \end_layout \begin_layout LyX-Code plot(tm,vel) \end_layout \begin_layout LyX-Code plot(tm,dis) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout LyX-Code \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none From the result we can see the phase relationship between the velocity and the displacement. If you want to see the effect of damping add that to the expression for force. Try changing parameters like spring constant, mass etc. to study the effect of them on the result. \end_layout \begin_layout Standard The above problem is solved using the Fourth order Runge-Kutta method in the program listed below. The displacement, as a function of time, calculated using RK method is compared with the result of the analytical expression. (spring2.py) \end_layout \begin_layout LyX-Code """ \end_layout \begin_layout LyX-Code The rk4_two() routine in this program does a two step integration using \end_layout \begin_layout LyX-Code an array method. The current x and xprime values are kept in a global \end_layout \begin_layout LyX-Code list named 'val'. \end_layout \begin_layout LyX-Code val[0] = current position; val[1] = current velocity \end_layout \begin_layout LyX-Code The results are compared with analytically calculated values. \end_layout \begin_layout LyX-Code """ \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code def accn(t, val): \end_layout \begin_layout LyX-Code force = -spring_const * val[0] - damping * val[1] \end_layout \begin_layout LyX-Code return force/mass \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def vel(t, val): \end_layout \begin_layout LyX-Code return val[1] \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def rk4_two(t, h): # Time and Step value \end_layout \begin_layout LyX-Code global xxp # x and xprime values in a 'xxp' \end_layout \begin_layout LyX-Code k1 = [0,0] # initialize 5 empty lists. \end_layout \begin_layout LyX-Code k2 = [0,0] \end_layout \begin_layout LyX-Code k3 = [0,0] \end_layout \begin_layout LyX-Code k4 = [0,0] \end_layout \begin_layout LyX-Code tmp= [0,0] \end_layout \begin_layout LyX-Code k1[0] = vel(t,xxp) \end_layout \begin_layout LyX-Code k1[1] = accn(t,xxp) \end_layout \begin_layout LyX-Code for i in range(2): # value of functions at t + h/2 \end_layout \begin_layout LyX-Code tmp[i] = xxp[i] + k1[i] * h/2 \end_layout \begin_layout LyX-Code k2[0] = vel(t + h/2, tmp) \end_layout \begin_layout LyX-Code k2[1] = accn(t + h/2, tmp) \end_layout \begin_layout LyX-Code for i in range(2): # value of functions at t + h/2 \end_layout \begin_layout LyX-Code tmp[i] = xxp[i] + k2[i] * h/2 \end_layout \begin_layout LyX-Code k3[0] = vel(t + h/2, tmp) \end_layout \begin_layout LyX-Code k3[1] = accn(t + h/2, tmp) \end_layout \begin_layout LyX-Code for i in range(2): # value of functions at t + h \end_layout \begin_layout LyX-Code tmp[i] = xxp[i] + k3[i] * h \end_layout \begin_layout LyX-Code k4[0] = vel(t+h, tmp) \end_layout \begin_layout LyX-Code k4[1] = accn(t+h, tmp) \end_layout \begin_layout LyX-Code for i in range(2): # value of functions at t + h \end_layout \begin_layout LyX-Code xxp[i] = xxp[i] + ( k1[i] + \backslash \end_layout \begin_layout LyX-Code 2.0*k2[i] + 2.0*k3[i] + k4[i]) * h/ 6.0 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code t = 0.0 # Stating time \end_layout \begin_layout LyX-Code h = 0.01 # Runge-Kutta step size, time increment \end_layout \begin_layout LyX-Code xxp = [2.0, 0.0] # initial position & velocity \end_layout \begin_layout LyX-Code spring_const = 100.0 # spring constant \end_layout \begin_layout LyX-Code mass = 2.0 # mass of the oscillating object \end_layout \begin_layout LyX-Code damping = 0.0 \end_layout \begin_layout LyX-Code tm = [0.0] # Lists to store time, position & velocity \end_layout \begin_layout LyX-Code x = [xxp[0]] \end_layout \begin_layout LyX-Code xp = [xxp[1]] \end_layout \begin_layout LyX-Code xth = [xxp[0]] \end_layout \begin_layout LyX-Code while t < 5: \end_layout \begin_layout LyX-Code rk4_two(t,h) # Do one step RK integration \end_layout \begin_layout LyX-Code t = t + h \end_layout \begin_layout LyX-Code tm.append(t) \end_layout \begin_layout LyX-Code xp.append(xxp[1]) \end_layout \begin_layout LyX-Code x.append(xxp[0]) \end_layout \begin_layout LyX-Code th = 2.0 * cos(sqrt(spring_const/mass)* (t)) \end_layout \begin_layout LyX-Code xth.append(th) \end_layout \begin_layout LyX-Code plot(tm,x) \end_layout \begin_layout LyX-Code plot(tm,xth,'+') \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Subsubsection Mass Spring problem in 3D \end_layout \begin_layout Standard The following program displays the movement of the mass attached toa spring in 3D graphics. (spring3d.py) \end_layout \begin_layout LyX-Code from visual import * \end_layout \begin_layout LyX-Code base = box (pos=(0,-1,0), length=16, height=0.1, width=4, color=color.blue) \end_layout \begin_layout LyX-Code wall = box (pos=(0,0,0), length=0.1, height=2, width=4, color=color.white) \end_layout \begin_layout LyX-Code ball = sphere (pos=(4,0,0), radius=1, color=color.red) \end_layout \begin_layout LyX-Code spring = helix(pos=(0,0,0), axis=(4,0,0), radius=0.5, color=color.red) \end_layout \begin_layout LyX-Code ball2 = sphere (pos=(-4,0,0), radius=1, color=color.green) \end_layout \begin_layout LyX-Code spring2 = helix(pos=(0,0,0), axis=(-4,0,0), radius=0.5, color=color.green) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code t = 0.0 \end_layout \begin_layout LyX-Code dt = 0.01 \end_layout \begin_layout LyX-Code x1 = 2.0 \end_layout \begin_layout LyX-Code x2 = -2.0 \end_layout \begin_layout LyX-Code v1 = 0.0 \end_layout \begin_layout LyX-Code v2 = 0.0 \end_layout \begin_layout LyX-Code k = 1000.0 \end_layout \begin_layout LyX-Code m = 1.0 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code while 1: \end_layout \begin_layout LyX-Code rate(20) \end_layout \begin_layout LyX-Code f1 = -k * x1 \end_layout \begin_layout LyX-Code v1 = v1 + (f1/m ) * dt # Acceleration = Force / mass ; dv = a.dt \end_layout \begin_layout LyX-Code f2 = -k * x2 - v2 # damping proportional to velocity \end_layout \begin_layout LyX-Code v2 = v2 + (f2/m ) * dt # Acceleration = Force / mass ; dv = a.dt \end_layout \begin_layout LyX-Code x1 = x1 + v1 * dt \end_layout \begin_layout LyX-Code x2 = x2 + v2 * dt \end_layout \begin_layout LyX-Code t = t + dt \end_layout \begin_layout LyX-Code spring.length = 4 + x1 \end_layout \begin_layout LyX-Code ball.x = x1 + 4 \end_layout \begin_layout LyX-Code spring2.length = 4 - x2 \end_layout \begin_layout LyX-Code ball2.x = x2 - 4 \end_layout \begin_layout Subsection Charged particle in Electric and Magnetic Fields \end_layout \begin_layout Standard The following example traces the movement of a charged particle under the influence of electric and magnetic fields. You can edit the program to change the components of the feilds to see their effect on the trajectory. For better results we should rewrite it using RK4 method. (cp_em.py) \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code import pylab as p \end_layout \begin_layout LyX-Code import matplotlib.axes3d as p3 \end_layout \begin_layout LyX-Code Ex = 0.0 # Components of Applied Electric Field \end_layout \begin_layout LyX-Code Ey = 2.0 \end_layout \begin_layout LyX-Code Ez = 0.0 \end_layout \begin_layout LyX-Code Bx = 0.0 # Magnetic field \end_layout \begin_layout LyX-Code By = 0.0 \end_layout \begin_layout LyX-Code Bz = 2.0 \end_layout \begin_layout LyX-Code m = 2.0 # Mass of the particle \end_layout \begin_layout LyX-Code q = 5.0 # Charge \end_layout \begin_layout LyX-Code x = 0.0 # Components of initial position and velocity \end_layout \begin_layout LyX-Code y = 0.0 \end_layout \begin_layout LyX-Code z = 0.0 \end_layout \begin_layout LyX-Code vx = 20.0 \end_layout \begin_layout LyX-Code vy = 0.0 \end_layout \begin_layout LyX-Code vz = 0.0 \end_layout \begin_layout LyX-Code a = [] \end_layout \begin_layout LyX-Code b = [] \end_layout \begin_layout LyX-Code c = [] \end_layout \begin_layout LyX-Code t = 0.0 \end_layout \begin_layout LyX-Code dt = 0.01 \end_layout \begin_layout LyX-Code while t < 6: # trace until time reaches 1 \end_layout \begin_layout LyX-Code Fx = q * (Ex + (vy * Bz) - (vz * By) ) \end_layout \begin_layout LyX-Code vx = vx + Fx/m * dt # Acceleration = F/m; dv = a.dt \end_layout \begin_layout LyX-Code Fy = q * (Ey - (vx * Bz) + (vz * Bx) ) \end_layout \begin_layout LyX-Code vy = vy + Fy/m * dt \end_layout \begin_layout LyX-Code Fz = q * (Ez + (vx * By) - (vy * Bx) ) \end_layout \begin_layout LyX-Code vz = vz + Fz/m * dt \end_layout \begin_layout LyX-Code x = x + vx * dt \end_layout \begin_layout LyX-Code y = y + vy * dt \end_layout \begin_layout LyX-Code z = z + vz * dt \end_layout \begin_layout LyX-Code a.append(x) \end_layout \begin_layout LyX-Code b.append(y) \end_layout \begin_layout LyX-Code c.append(z) \end_layout \begin_layout LyX-Code t = t + dt \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code p.plot(c,b) \end_layout \begin_layout LyX-Code fig=p.figure() \end_layout \begin_layout LyX-Code ax = p3.Axes3D(fig) \end_layout \begin_layout LyX-Code ax.set_xlabel('X') \end_layout \begin_layout LyX-Code ax.set_ylabel('Y') \end_layout \begin_layout LyX-Code ax.set_zlabel('Z') \end_layout \begin_layout LyX-Code ax.plot3D(a,b,c) \end_layout \begin_layout LyX-Code p.show() \end_layout \begin_layout LyX-Code \end_layout \begin_layout Subsection Motion in a central field \end_layout \begin_layout Standard Run the program cf3d.py from the CD to simulate motion of a mass in a gravitation al field. \end_layout \begin_layout Section Matrix manipulation \end_layout \begin_layout Standard (todo) \end_layout \begin_layout Section Random Distributions \end_layout \begin_layout Standard (todo) \end_layout \end_body \end_document pycode-browser-1.02/Code/Physics/readme000077500000000000000000000001711254404601000200520ustar00rootroot00000000000000This directory contains small examples of numerical computation using Python. To run them, for example; #python plot1.pypycode-browser-1.02/Code/Physics/rk_sin.py000077500000000000000000000011311254404601000205260ustar00rootroot00000000000000from pylab import * def func(t): return cos(t) def rk4(s,t): k1 = dt * func(t) k2 = dt * func(t + dt/2.0) k3 = dt * func(t + dt/2.0) k4 = dt * func(t + dt) return s + ( k1/6 + k2/3 + k3/3 + k4/6 ) t = 0.0 # Stating angle dt = 0.01 # value of angle increment val = 0.0 # value of the 'sine' function at t = 0 tm = [0.0] # List to store theta values res = [0.0] # RK4 results while t < 2*pi: val = rk4(val,t) # get the new value t = t + dt tm.append(t) res.append(val) plot(tm, res,'+') plot(tm, sin(tm)) # compare with actual curve show() pycode-browser-1.02/Code/Physics/series_cos.py000077500000000000000000000005041254404601000214020ustar00rootroot00000000000000import scipy, pylab def mycos(x): res = 0.0 for n in range(18): res = res + (-1)**n * (x ** (2*n)) / scipy.factorial(2*n) return res def vmycos(ax): y = [] for x in ax: y.append(mycos(x)) return y x = scipy.linspace(0,4*scipy.pi,100) y = vmycos(x) pylab.plot(x,y) pylab.plot(x,scipy.cos(x),'+') pylab.show() pycode-browser-1.02/Code/Physics/series_sin.py000077500000000000000000000003651254404601000214140ustar00rootroot00000000000000from pylab import * from scipy import * x = linspace(-pi,pi,40) a = zeros(40) plot(x,sin(x)) for n in range(1,5): sign = (-1)**(n+1) term = x**(2*n-1) / factorial(2*n-1) a = a + sign * term print n,sign plot(x,term) plot(x,a,'+') show() pycode-browser-1.02/Code/Physics/sph1.py000077500000000000000000000006671254404601000201310ustar00rootroot00000000000000from pylab import * from scipy import special a_th = [] # list to store polar angle theta from -90 to + 90 deg a_sph = [] # list to store absolute values if sperical harminics phi = 0.0 # Fix azimuth, phi at zero theta = -pi/2 # start theta from -90 deg while theta < pi/2: h = special.sph_harm(0,10, phi, theta) # (m, l , phi, theta) a_sph.append(abs(h)) a_th.append(theta * 180/pi) theta = theta + 0.02 plot(a_th,a_sph) show() pycode-browser-1.02/Code/Physics/sph2.py000077500000000000000000000012051254404601000201170ustar00rootroot00000000000000from numpy import * from scipy import special import pylab as p import matplotlib.axes3d as p3 phi = linspace(0, 2*pi, 50) theta = linspace(-pi/2, pi/2, 200) ax = [] ay = [] az = [] R = 1.0 for t in theta: polar = float(t) for k in phi: azim = float(k) sph = special.sph_harm(0,2,azim, polar) # Y(m,l,phi,theta) modulation = 0.2 * abs(sph) r = R * ( 1 + modulation) x = r*cos(polar)*cos(azim) y = r*cos(polar)*sin(azim) z = r*sin(polar) # print z # print x,y,z ax.append(x) ay.append(y) az.append(z) fig=p.figure() f = p3.Axes3D(fig) f.set_xlabel('X') f.set_ylabel('Y') f.set_zlabel('Z') f.scatter3D(ax,ay,az) p.show() pycode-browser-1.02/Code/Physics/spring1.py000077500000000000000000000011511254404601000206260ustar00rootroot00000000000000#spring1.py from pylab import * t = 0.0 # Stating time dt = 0.01 # value of time increment x = 10.0 # initial position v = 0.0 # initial velocity k = 10.0 # spring constant m = 2.0 # mass of the oscillating object tm = [] # Empty lists to store time, velocity and displacement vel = [] dis = [] while t < 5: f = -k * x # Try (-k*x - 0.5 * v) to add damping v = v + (f/m ) * dt # dv = a.dt x = x + v * dt # dS = v.dt t = t + dt tm.append(t) vel.append(v) dis.append(x) plot(tm,vel) plot(tm,dis) show() pycode-browser-1.02/Code/Physics/spring2.py000077500000000000000000000033701254404601000206340ustar00rootroot00000000000000""" spring2.py The rk4_two() routine in this program does a two step integration using an array method. The current x and xprime values are kept in a global list named 'val'. val[0] = current position; val[1] = current velocity The results are compared with analytically calculated values. """ from pylab import * def accn(t, val): force = -spring_const * val[0] - damping * val[1] return force/mass def vel(t, val): return val[1] def rk4_two(t, h): # Time and Step value global xxp # x and xprime values in a 'xxp' k1 = [0,0] # initialize 5 empty lists. k2 = [0,0] k3 = [0,0] k4 = [0,0] tmp= [0,0] k1[0] = vel(t,xxp) k1[1] = accn(t,xxp) for i in range(2): # value of functions at t + h/2 tmp[i] = xxp[i] + k1[i] * h/2 k2[0] = vel(t + h/2, tmp) k2[1] = accn(t + h/2, tmp) for i in range(2): # value of functions at t + h/2 tmp[i] = xxp[i] + k2[i] * h/2 k3[0] = vel(t + h/2, tmp) k3[1] = accn(t + h/2, tmp) for i in range(2): # value of functions at t + h tmp[i] = xxp[i] + k3[i] * h k4[0] = vel(t+h, tmp) k4[1] = accn(t+h, tmp) for i in range(2): # value of functions at t + h xxp[i] = xxp[i] + ( k1[i] + \ 2.0*k2[i] + 2.0*k3[i] + k4[i]) * h/ 6.0 t = 0.0 # Stating time h = 0.01 # Runge-Kutta step size, time increment xxp = [2.0, 0.0] # initial position & velocity spring_const = 100.0 # spring constant mass = 2.0 # mass of the oscillating object damping = 0.0 tm = [0.0] # Lists to store time, position & velocity x = [xxp[0]] xp = [xxp[1]] xth = [xxp[0]] while t < 5: rk4_two(t,h) # Do one step RK integration t = t + h tm.append(t) xp.append(xxp[1]) x.append(xxp[0]) th = 2.0 * cos(sqrt(spring_const/mass)* (t)) xth.append(th) plot(tm,x) plot(tm,xth,'+') show() pycode-browser-1.02/Code/Physics/spring3d.py000077500000000000000000000016061254404601000210010ustar00rootroot00000000000000#spring3d.py from visual import * base = box (pos=(0,-1,0), length=16, height=0.1, width=4, color=color.blue) wall = box (pos=(0,0,0), length=0.1, height=2, width=4, color=color.white) ball = sphere (pos=(4,0,0), radius=1, color=color.red) spring = helix(pos=(0,0,0), axis=(4,0,0), radius=0.5, color=color.red) ball2 = sphere (pos=(-4,0,0), radius=1, color=color.green) spring2 = helix(pos=(0,0,0), axis=(-4,0,0), radius=0.5, color=color.green) t = 0.0 dt = 0.01 x1 = 2.0 x2 = -2.0 v1 = 0.0 v2 = 0.0 k = 1000.0 m = 1.0 while 1: rate(20) f1 = -k * x1 v1 = v1 + (f1/m ) * dt # Acceleration = Force / mass ; dv = a.dt f2 = -k * x2 - v2 # damping proportional to velocity v2 = v2 + (f2/m ) * dt # Acceleration = Force / mass ; dv = a.dt x1 = x1 + v1 * dt x2 = x2 + v2 * dt t = t + dt spring.length = 4 + x1 ball.x = x1 + 4 spring2.length = 4 - x2 ball2.x = x2 - 4 pycode-browser-1.02/Code/PythonBook/000077500000000000000000000000001254404601000173425ustar00rootroot00000000000000pycode-browser-1.02/Code/PythonBook/chap2/000077500000000000000000000000001254404601000203375ustar00rootroot00000000000000pycode-browser-1.02/Code/PythonBook/chap2/area.py000077500000000000000000000001601254404601000216210ustar00rootroot00000000000000#Example: area.py pi = 3.1416 r = input('Enter Radius ') a = pi * r ** 2 #A=\pi r^{2} print 'Area = ', a pycode-browser-1.02/Code/PythonBook/chap2/badtable.py000077500000000000000000000001231254404601000224460ustar00rootroot00000000000000#Example: badtable.py print 1 * 8 print 2 * 8 print 3 * 8 print 4 * 8 print 5 * 8 pycode-browser-1.02/Code/PythonBook/chap2/big.py000077500000000000000000000002331254404601000214530ustar00rootroot00000000000000#Example: big.py x = input('Enter a number ') if x > 10: print 'Bigger Number' elif x < 10: print 'Smaller Number' else: print 'Same Number' pycode-browser-1.02/Code/PythonBook/chap2/big2.py000077500000000000000000000002141254404601000215340ustar00rootroot00000000000000#Example: big2.py x = 1 while x < 11: if x < 5: print 'Samll ', x else: print 'Big ', x x = x + 1 print 'Done' pycode-browser-1.02/Code/PythonBook/chap2/big3.py000077500000000000000000000003021254404601000215330ustar00rootroot00000000000000x = 1 while x < 10: if x < 3: print 'skipping work', x x = x + 1 continue print x if x == 4: print 'Enough of work' break x = x + 1 print 'Done' pycode-browser-1.02/Code/PythonBook/chap2/compare.py000077500000000000000000000001201254404601000223330ustar00rootroot00000000000000x = raw_input('Enter a string ') if x == 'hello': print 'You typed ', x pycode-browser-1.02/Code/PythonBook/chap2/conflict.py000077500000000000000000000003651254404601000225210ustar00rootroot00000000000000#Example conflict.py from numpy import * x = linspace(0, 5, 10) # make a 10 element array print sin(x) # sin of scipy can handle arrays from math import * # sin of math will be called now print sin(x) # will give ERROR pycode-browser-1.02/Code/PythonBook/chap2/cpoint.py000077500000000000000000000004301254404601000222050ustar00rootroot00000000000000from point import * class colPoint(Point): #colPoint inherits Point color = 'black' def __init__(self,x=0, y=0, col='black'): Point.__init__(self,x,y) self.color = col def __str__(self): return '%s colored Point at (%f,%f)'%(self.color,self.xpos, self.ypos) pycode-browser-1.02/Code/PythonBook/chap2/draw.py000077500000000000000000000122321254404601000216510ustar00rootroot00000000000000from Tkinter import * AXWIDTH = 30 # width of the axis display canvas class disp: """ Class for displaying items in a canvas using a world coordinate system. The range of the world coordinate system is specified by calling the setWorld method. """ border = 0 pad = 0 bordcol = 'grey' # Border color gridcol = 'grey' # Grid color bgcolor = '#dbdbdb' # background color for all plotbg = 'ivory' # Plot window background color textcolor = 'blue' traces = [] xtext = [] ytext = [] markerval = [] markertext = None def __init__(self, parent, width=400., height=100.,color = 'ivory'): self.parent = parent self.SCX = width self.SCY = height self.plotbg = color f = Frame(parent, bg = self.bgcolor, borderwidth = self.pad) f.pack(side=TOP) self.yaxis = Canvas(f, width = AXWIDTH, height = height, bg = self.bgcolor) self.yaxis.pack(side = LEFT, anchor = N, pady = self.border) f1 = Frame(f) f1.pack(side=LEFT) self.canvas = Canvas(f1, background = self.plotbg, width = width, height = height) self.canvas.pack(side = TOP) self.canvas.bind("", self.show_xy) self.xaxis = Canvas(f1, width = width, height = AXWIDTH, bg = self.bgcolor) self.xaxis.pack(side = LEFT, anchor = N, padx = self.border) self.canvas.create_rectangle ([(1,1),(width,height)], outline = self.bordcol) self.canvas.pack() Canvas(f, width = AXWIDTH, height = height, bg = self.bgcolor).pack(side=LEFT) # spacer only self.setWorld(0 , 0, self.SCX, self.SCY) # initialize scale factors self.grid(10,100) def setWorld(self, x1, y1, x2, y2): #Calculates the scale factors self.xmin = float(x1) self.ymin = float(y1) self.xmax = float(x2) self.ymax = float(y2) self.xscale = (self.xmax - self.xmin) / (self.SCX) self.yscale = (self.ymax - self.ymin) / (self.SCY) def w2s(self, p): # World to Screen xy conversion before plotting anything ip = [] for xy in p: ix = self.border + int( (xy[0] - self.xmin) / self.xscale) iy = self.border + int( (xy[1] - self.ymin) / self.yscale) iy = self.SCY - iy ip.append((ix,iy)) return ip def show_xy(self,event): #Prints the XY coordinates of the current cursor position ix = self.canvas.canvasx(event.x) - self.border iy = self.SCY - self.canvas.canvasy(event.y) #- self.border x = ix * self.xscale + self.xmin y = iy * self.yscale + self.ymin s = 'x = %5.3f\ny = %5.3f' % (x,y) try: self.canvas.delete(self.markertext) except: pass self.markertext = self.canvas.create_text(self.border + 1,\ self.SCY-1, anchor = SW, justify = LEFT, text = s) self.markerval = [x,y] def mark_axes(self, xlab='milli seconds', ylab='mV', numchans=1): # Draw the axis labels and values numchans = 1 for t in self.xtext: # display after dividing by scale factors self.xaxis.delete(t) for t in self.ytext: self.yaxis.delete(t) self.xtext = [] self.ytext = [] dx = float(self.SCX)/5 for x in range(0,6): a = numchans * x *(self.xmax - self.xmin)/5 + self.xmin s = '%4.1f'%(a) adjust = 0 if x == 0: adjust = 6 if x == 5: adjust = -10 t = self.xaxis.create_text(int(x*dx)+adjust,1,text = s, anchor=N, fill = self.textcolor) self.xtext.append(t) self.xtext.append(self.xaxis.create_text(int(self.SCX/2)\ ,AXWIDTH, text = xlab, anchor=S, fill = self.textcolor)) dy = float(self.SCY)/5 for y in range(0,6): a = y*(self.ymax - self.ymin)/5 # + self.ymin if self.ymax > 99: s = '%4.0f'%(self.ymax-a) else: s = '%4.1f'%(self.ymax-a) adjust = 0 if y == 0: adjust = 6 if y == 5: adjust = -5 t = self.yaxis.create_text(AXWIDTH, int(y*dy)+adjust, \ text = s,anchor = E, fill = self.textcolor) self.ytext.append(t) self.ytext.append(self.yaxis.create_text(0,self.SCY/2,\ text = ylab, anchor=W, fill = self.textcolor)) def box(self, x1, y1, x2, y2, col): ip = self.w2s([(x1,y1),(x2,y2)]) self.canvas.create_rectangle(ip, outline=col) def line(self, points, col, permanent = False, smooth = True): ip = self.w2s(points) t = self.canvas.create_line(ip, fill=col, smooth = smooth) if permanent == False: self.traces.append(t) def delete_lines(self): for t in self.traces: self.canvas.delete(t) self.traces = [] def grid(self, major, minor): dx = (self.xmax - self.xmin) / major dy = (self.ymax - self.ymin) / major x = self.xmin + dx while x < self.xmax: self.line([(x,self.ymin),(x,self.ymax)],self.gridcol, True) x = x +dx y = self.ymin + dy while y < self.ymax: self.line([(self.xmin,y),(self.xmax,y)],self.gridcol, True) y = y +dy #------------------------------- disp class end -------------------------------------- root = Tk() for k in range(3): w = disp(root, width=400, height=100) w.setWorld(0, 0, 100, 100) w.mark_axes('X','Y') root.mainloop() pycode-browser-1.02/Code/PythonBook/chap2/except.py000077500000000000000000000001441254404601000222030ustar00rootroot00000000000000x = input('Enter a number ') try: print 10.0/x except: print 'Division by zero not allowed' pycode-browser-1.02/Code/PythonBook/chap2/except2.py000077500000000000000000000003271254404601000222700ustar00rootroot00000000000000import string def get_number(): while 1: try: a = raw_input('Enter a number ') x = string.atof(a) return x except: print 'Enter a valid number' print get_number() pycode-browser-1.02/Code/PythonBook/chap2/factor.py000077500000000000000000000002441254404601000221720ustar00rootroot00000000000000#Example factor.py def factorial(n): # a recursive function if n == 0: return 1 else: return n * factorial(n-1) print factorial(10) pycode-browser-1.02/Code/PythonBook/chap2/fibanocci.py000077500000000000000000000002001254404601000226210ustar00rootroot00000000000000def fib(n): # write Fibonacci series up to n a, b = 0, 1 while b < n: print b, a, b = b, a+b fib(30)pycode-browser-1.02/Code/PythonBook/chap2/first.py000077500000000000000000000005441254404601000220460ustar00rootroot00000000000000 #Example: first.py ''' This is Python program starting with a multi-line comment, enclosed within triple quotes. In a single line, anything after a # sign is a comment ''' x = 10 print x, type(x) #print x and its type y = 10.4 print y, type(y) z = 3 + 4j print z, type(z) s1 = 'I am a String ' s2 = 'me too' print s1, s2, type(s1) pycode-browser-1.02/Code/PythonBook/chap2/forloop.py000077500000000000000000000001741254404601000223760ustar00rootroot00000000000000#Example: forloop.py a = 'my name' for ch in a: print ch b = ['hello', 3.4, 2345, 3+5j] for item in b: print item pycode-browser-1.02/Code/PythonBook/chap2/forloop2.py000077500000000000000000000001321254404601000224520ustar00rootroot00000000000000#Example: forloop2.py mylist = range(5) print mylist for item in mylist: print item pycode-browser-1.02/Code/PythonBook/chap2/forloop3.py000077500000000000000000000001231254404601000224530ustar00rootroot00000000000000#Example: forloop3.py mylist = range(5,51,5) for item in mylist: print item , pycode-browser-1.02/Code/PythonBook/chap2/forloop4.py000077500000000000000000000001351254404601000224570ustar00rootroot00000000000000a = [2, 5, 3, 4, 12] size = len(a) for k in range(size): if a[k] < 0: a[k] = 0 print a pycode-browser-1.02/Code/PythonBook/chap2/format.py000077500000000000000000000001301254404601000221760ustar00rootroot00000000000000#Example: format.py a = 2.0 /3 print a print 'a = %5.3f' %(a) # upto 3 decimal places pycode-browser-1.02/Code/PythonBook/chap2/format2.py000077500000000000000000000003141254404601000222640ustar00rootroot00000000000000#Example: format2.py a = 'justify as you like' print '%30s'%a print '%-30s'%a # minus sign for left justification for k in range(1,11): # A good looking table print '5 x %2d = %2d' %(k, k*5)pycode-browser-1.02/Code/PythonBook/chap2/func.py000077500000000000000000000001321254404601000216430ustar00rootroot00000000000000#Example func.py def sum(a,b): # a trivial function return a + b print sum(3, 4) pycode-browser-1.02/Code/PythonBook/chap2/global.py000077500000000000000000000001641254404601000221550ustar00rootroot00000000000000def change(x): global counter # use the global variable counter = x counter = 10 change(5) print counterpycode-browser-1.02/Code/PythonBook/chap2/hello.py000077500000000000000000000000241254404601000220130ustar00rootroot00000000000000print 'Hello World' pycode-browser-1.02/Code/PythonBook/chap2/kin1.py000077500000000000000000000002201254404601000215500ustar00rootroot00000000000000x = input('Enter an integer ') y = input('Enter one more ') print 'The sum is ', x + y s = raw_input('Enter a String ') print 'You entered ', s pycode-browser-1.02/Code/PythonBook/chap2/kin2.py000077500000000000000000000003201254404601000215520ustar00rootroot00000000000000x,y = input('Enter x and y separated by comma ') print 'The sum is ', x + y s = raw_input('Enter a decimal number ') a = float(s) print s * 2 # prints string twice print a * 2 # converted value times 2 pycode-browser-1.02/Code/PythonBook/chap2/list1.py000077500000000000000000000001351254404601000217470ustar00rootroot00000000000000a = [3, 3.5, 234] # make a list print a[0] a[1] = 'haha' # Change an element print a pycode-browser-1.02/Code/PythonBook/chap2/list2.py000077500000000000000000000000761254404601000217540ustar00rootroot00000000000000a = [1,2] print a * 2 print a + [3,4] b = [10, 20, a] print b pycode-browser-1.02/Code/PythonBook/chap2/list3.py000077500000000000000000000002161254404601000217510ustar00rootroot00000000000000a = [] # make an empty list a.append(3) # Add an element a.insert(0,2.3) # insert 2.3 as first element print a, a[0] print len(a)pycode-browser-1.02/Code/PythonBook/chap2/list_copy.py000077500000000000000000000001741254404601000227230ustar00rootroot00000000000000a = [1,2,3,4] print a b = a print a == b b[0] = 5 print a import copy c = copy.copy(a) c[1] = 100 print a is c print a, c pycode-browser-1.02/Code/PythonBook/chap2/mathlocal.py000077500000000000000000000001321254404601000226540ustar00rootroot00000000000000#Example mathlocal.py from math import sin # sin is imported as local print sin(0.5) pycode-browser-1.02/Code/PythonBook/chap2/mathlocal2.py000077500000000000000000000001331254404601000227370ustar00rootroot00000000000000#Example mathlocal2.py from math import * # import everything from math print sin(0.5) pycode-browser-1.02/Code/PythonBook/chap2/mathsin.py000077500000000000000000000001161254404601000223550ustar00rootroot00000000000000#Example math.py import math print math.sin(0.5) # module_name.method_name pycode-browser-1.02/Code/PythonBook/chap2/mathsin2.py000077500000000000000000000001541254404601000224410ustar00rootroot00000000000000#Example math2.py import math as m # Give another madule name print m.sin(0.5) # Refer by the new name pycode-browser-1.02/Code/PythonBook/chap2/max.py000077500000000000000000000002421254404601000214770ustar00rootroot00000000000000#Example: max.py max = 0 while 1: # Infinite loop x = input('Enter a number ') if x > max: max = x if x == 0: print max break pycode-browser-1.02/Code/PythonBook/chap2/mutable.py000077500000000000000000000002331254404601000223430ustar00rootroot00000000000000s = [3, 3.5, 234] # make a list s[2] = 'haha' # Change an element print s x = 'myname' # String type #x[1] = 2 # Will give ERROR pycode-browser-1.02/Code/PythonBook/chap2/named.py000077500000000000000000000002131254404601000217740ustar00rootroot00000000000000def power(mant = 10.0, exp = 2.0): return mant ** exp print power(5., 3) print power(4.) # prints 16 print power(exp=3) pycode-browser-1.02/Code/PythonBook/chap2/oper.py000077500000000000000000000001401254404601000216540ustar00rootroot00000000000000x = 2 y = 4 print x + y * 2 s = 'Hello ' print s + s print 3 * s print x == y print y == 2 * x pycode-browser-1.02/Code/PythonBook/chap2/pickledump.py000077500000000000000000000001611254404601000230470ustar00rootroot00000000000000#Example pickle.py import pickle f = open('test.pck' , 'w') pickle.dump(12.3, f) # write a float type f.close() pycode-browser-1.02/Code/PythonBook/chap2/pickleload.py000077500000000000000000000002221254404601000230170ustar00rootroot00000000000000#Example pickle2.py import pickle f = open('test.pck' , 'r') x = pickle.load(f) print x , type(x) # check the type of data read f.close() pycode-browser-1.02/Code/PythonBook/chap2/point.py000077500000000000000000000010441254404601000220440ustar00rootroot00000000000000class Point: xpos = 0 ypos = 0 def __init__(self, x=0, y=0): self.xpos = x self.ypos = y def __str__(self): return 'Point at (%f,%f)'%(self.xpos, self.ypos) def __add__(self, other): xpos = self.xpos + other.xpos ypos = self.ypos + other.ypos res = Point(xpos,ypos) return res def __sub__(self, other): import math dx = self.xpos - other.xpos dy = self.ypos - other.ypos return math.sqrt(dx**2+dy**2) def dist(self): import math return math.sqrt(self.xpos**2 + self.ypos**2) pycode-browser-1.02/Code/PythonBook/chap2/point1.py000077500000000000000000000002221254404601000221220ustar00rootroot00000000000000from point import * origin = Point() print origin p1 = Point(4,4) p2 = Point(8,7) print p1 print p2 print p1 + p2 print p1 - p2 print p1.dist() pycode-browser-1.02/Code/PythonBook/chap2/point2.py000077500000000000000000000001651254404601000221310ustar00rootroot00000000000000from cpoint import * p1 = Point(5,5) rp1 = colPoint(2,2,'red') print p1 print rp1 print rp1 + p1 print rp1.dist() pycode-browser-1.02/Code/PythonBook/chap2/power.py000077500000000000000000000001331254404601000220450ustar00rootroot00000000000000def power(mant, exp = 2.0): return mant ** exp print power(5., 3) print power(4.) pycode-browser-1.02/Code/PythonBook/chap2/reverse.py000077500000000000000000000001031254404601000223610ustar00rootroot00000000000000a = 'abcd' b = '' for k in range(len(a)): b = b + a[-1-k] print b pycode-browser-1.02/Code/PythonBook/chap2/rfile.py000077500000000000000000000001071254404601000220130ustar00rootroot00000000000000#Example rfile.py f = open('test.dat' , 'r') print f.read() f.close() pycode-browser-1.02/Code/PythonBook/chap2/rfile2.py000077500000000000000000000002311254404601000220730ustar00rootroot00000000000000#Example rfile2.py f = open('test.dat' , 'r') print f.read(7) # get first seven characters print f.read() # get the remaining ones f.close() pycode-browser-1.02/Code/PythonBook/chap2/rfile3.py000077500000000000000000000004041254404601000220760ustar00rootroot00000000000000#Example rfile3.py f = open('data.dat' , 'r') while 1: # infinite loop s = f.readline() if s == '' : # Empty string means end of file break # terminate the loop m = int(s) # Convert to integer print m * 5 f.close() pycode-browser-1.02/Code/PythonBook/chap2/scope.py000077500000000000000000000001101254404601000220150ustar00rootroot00000000000000def change(x): counter = x counter = 10 change(5) print counterpycode-browser-1.02/Code/PythonBook/chap2/slice.py000077500000000000000000000000671254404601000220160ustar00rootroot00000000000000a = 'hello world' print a[3:5] print a[6:] print a[:5] pycode-browser-1.02/Code/PythonBook/chap2/split.py000077500000000000000000000001601254404601000220440ustar00rootroot00000000000000s = 'I am a long string' print s.split() a = 'abc.abc.abc' aa = a.split('.') print aa mm = '+'.join(aa) print mmpycode-browser-1.02/Code/PythonBook/chap2/string.py000077500000000000000000000002311254404601000222160ustar00rootroot00000000000000s = 'hello world' print s[0] # print first element, h print s[1] # print e print s[-1] # will print the last characterpycode-browser-1.02/Code/PythonBook/chap2/string.pyc000077500000000000000000000003301254404601000223610ustar00rootroot00000000000000 Z-Lc@s%dZedGHedGHedGHdS(s hello worldiiiN(ts(((s5/home/phoenix/Desktop/PythonBook/code/chap2/string.pyts  pycode-browser-1.02/Code/PythonBook/chap2/string2.py000077500000000000000000000001031254404601000222760ustar00rootroot00000000000000a = 'hello'+'world' print a b = 'ha' * 3 print b print a[-1] + b[0]pycode-browser-1.02/Code/PythonBook/chap2/stringhelp.py000077500000000000000000000001301254404601000230650ustar00rootroot00000000000000s = 'hello world' print len(s) print s.upper() help(str) # press q to quit helppycode-browser-1.02/Code/PythonBook/chap2/submodule.py000077500000000000000000000001341254404601000227110ustar00rootroot00000000000000from numpy import * print random.normal() import scipy.special print scipy.special.j0(.1) pycode-browser-1.02/Code/PythonBook/chap2/table.py000077500000000000000000000001071254404601000220010ustar00rootroot00000000000000#Example: table.py x = 1 while x <= 10: print x * 8 x = x + 1 pycode-browser-1.02/Code/PythonBook/chap2/tkbutton.py000077500000000000000000000003061254404601000225650ustar00rootroot00000000000000from Tkinter import * def hello(): print 'hello world' w = Tk() # Creates the main Graphics window b = Button(w, text = 'Click Me', command = hello) b.pack() w.mainloop()pycode-browser-1.02/Code/PythonBook/chap2/tkcanvas.py000077500000000000000000000003131254404601000225230ustar00rootroot00000000000000from Tkinter import * def draw(event): c.create_rectangle(event.x, event.y, event.x+5, event.y+5) w = Tk() c = Canvas(w, width = 300, height = 200) c.pack() c.bind("", draw) w.mainloop() pycode-browser-1.02/Code/PythonBook/chap2/tkcanvas2.py000077500000000000000000000007171254404601000226150ustar00rootroot00000000000000from Tkinter import * recs = [] # List keeping track of the rectangles def remove(event): global recs if len(recs) > 0: c.delete(recs[0]) # delete from Canvas recs.pop(0) # delete first item from list def draw(event): r = c.create_rectangle(event.x, event.y, event.x+5, event.y+5) recs.append(r) w = Tk() c = Canvas(w, width = 300, height = 200) c.pack() c.bind("", draw) c.bind("", remove) w.mainloop() pycode-browser-1.02/Code/PythonBook/chap2/tklabel.py000077500000000000000000000001411254404601000223260ustar00rootroot00000000000000from Tkinter import * root = Tk() w = Label(root, text="Hello, world") w.pack() root.mainloop() pycode-browser-1.02/Code/PythonBook/chap2/tkmain.py000077500000000000000000000000621254404601000221750ustar00rootroot00000000000000from Tkinter import * root = Tk() root.mainloop()pycode-browser-1.02/Code/PythonBook/chap2/tkplot.py000077500000000000000000000004201254404601000222250ustar00rootroot00000000000000from tkplot_class import * from math import * w = Tk() gw1 = disp(w) xy = [] for k in range(200): x = 2 * pi * k/200 y = sin(x) xy.append((x,y)) gw1.setWorld(0, -1.0, 2*pi, 1.0) gw1.line(xy) gw2 = disp(w) gw2.line([(10,10),(100,100),(350,50)], 'red') w.mainloop() pycode-browser-1.02/Code/PythonBook/chap2/tkplot_class.py000077500000000000000000000025471254404601000234260ustar00rootroot00000000000000from Tkinter import * from math import * class disp: """ Class for displaying items in a canvas using a world coordinate system. The range of the world coordinate system is specified by calling the setWorld method. """ traces = [] def __init__(self, parent, width=400., height=200.): self.parent = parent self.SCX = width self.SCY = height self.border = 1 self.canvas = Canvas(parent, width=width, height=height) self.canvas.pack(side = LEFT) self.setWorld(0 , 0, self.SCX, self.SCY) # initialize scale factors def setWorld(self, x1, y1, x2, y2): #Calculates the scale factors self.xmin = float(x1) self.ymin = float(y1) self.xmax = float(x2) self.ymax = float(y2) self.xscale = (self.xmax - self.xmin) / (self.SCX) self.yscale = (self.ymax - self.ymin) / (self.SCY) def w2s(self, p): # World to Screen xy conversion before plotting anything ip = [] for xy in p: ix = self.border + int( (xy[0] - self.xmin) / self.xscale) iy = self.border + int( (xy[1] - self.ymin) / self.yscale) iy = self.SCY - iy ip.append((ix,iy)) return ip def line(self, points, col='blue'): ip = self.w2s(points) t = self.canvas.create_line(ip, fill=col) self.traces.append(t) def delete_lines(self): for t in self.traces: self.canvas.delete(t) self.traces = [] pycode-browser-1.02/Code/PythonBook/chap2/turtle1.py000077500000000000000000000003571254404601000223210ustar00rootroot00000000000000from turtle import * a = Pen() # Creates a graphics window a.forward(50) a.left(45) a.backward(50) a.right(45) a.forward(50) a.circle(10) a.up() a.forward(50) a.down() a.color('red') a.right(90) a.forward(50) raw_input('Press Enter') pycode-browser-1.02/Code/PythonBook/chap2/turtle2.py000077500000000000000000000002071254404601000223140ustar00rootroot00000000000000from turtle import * a = Pen() # Creates a graphics window for k in range(4): a.forward(50) a.left(90) a.circle(25) raw_input()pycode-browser-1.02/Code/PythonBook/chap2/turtle3.py000077500000000000000000000003261254404601000223170ustar00rootroot00000000000000from turtle import * def draw_rectangle(): for k in range(4): a.forward(50) a.left(90) a = Pen() # Creates a graphics window for k in range(36): draw_rectangle() a.left(10) raw_input()pycode-browser-1.02/Code/PythonBook/chap2/turtle4.py000077500000000000000000000004271254404601000223220ustar00rootroot00000000000000from turtle import * def f(length, depth): if depth == 0: forward(length) else: f(length/3, depth-1) right(60) f(length/3, depth-1) left(120) f(length/3, depth-1) right(60) f(length/3, depth-1) f(500, 4) raw_input('Press any Key') pycode-browser-1.02/Code/PythonBook/chap2/wfile.py000077500000000000000000000001301254404601000220140ustar00rootroot00000000000000#Example wfile.py f = open('test.dat' , 'w') f.write ('This is a test file') f.close() pycode-browser-1.02/Code/PythonBook/chap2/wfile2.py000077500000000000000000000001671254404601000221100ustar00rootroot00000000000000#Example wfile2.py f = open('data.dat' , 'w') for k in range(1,4): s = '%3d\n' %(k) f.write(s) f.close() pycode-browser-1.02/Code/PythonBook/chap3/000077500000000000000000000000001254404601000203405ustar00rootroot00000000000000pycode-browser-1.02/Code/PythonBook/chap3/aroper.py000077500000000000000000000001621254404601000222040ustar00rootroot00000000000000#Example oper.py from numpy import * a = array([[2,3], [4,5]]) b = array([[1,2], [3,0]]) print a + b print a * b pycode-browser-1.02/Code/PythonBook/chap3/array_copy.py000077500000000000000000000001451254404601000230650ustar00rootroot00000000000000from numpy import * a = zeros(3) print a b = a c = a.copy() c[0] = 10 print a, c b[0] = 10 print a,c pycode-browser-1.02/Code/PythonBook/chap3/cross.py000077500000000000000000000001441254404601000220450ustar00rootroot00000000000000#Example cross.py from numpy import * a = array([1,2,3]) b = array([4,5,6]) c = cross(a,b) print c pycode-browser-1.02/Code/PythonBook/chap3/fileio.py000077500000000000000000000002061254404601000221620ustar00rootroot00000000000000#Example fileio.py from numpy import * a = arange(10) print a a.tofile('myfile.dat') b = fromfile('myfile.dat', dtype='int') print b pycode-browser-1.02/Code/PythonBook/chap3/inv.py000077500000000000000000000002231254404601000215060ustar00rootroot00000000000000#Example inv.py from numpy import * a = array([ [4,1,-2], [2,-3,3], [-6,-2,1] ], dtype='float') ainv = linalg.inv(a) print ainv print dot(a,ainv) pycode-browser-1.02/Code/PythonBook/chap3/numpy1.py000077500000000000000000000001461254404601000221470ustar00rootroot00000000000000#Example numpy1.py from numpy import * x = array( [1,2] ) # Make array from list print x , type(x) pycode-browser-1.02/Code/PythonBook/chap3/numpy2.py000077500000000000000000000003131254404601000221440ustar00rootroot00000000000000#Example numpy2.py from numpy import * a = arange(1.0, 2.0, 0.1) # start, stop & step print a b = linspace(1,2,11) print b c = ones(5) print c d = zeros(5) print d e = random.rand(5) print e pycode-browser-1.02/Code/PythonBook/chap3/numpy3.py000077500000000000000000000002071254404601000221470ustar00rootroot00000000000000#Example numpy3.py from numpy import * a = [ [1,2] , [3,4] ] # make a list of lists x = array(a) # and convert to an array print a pycode-browser-1.02/Code/PythonBook/chap3/numpy4.py000077500000000000000000000001611254404601000221470ustar00rootroot00000000000000#Example numpy4.py from numpy import * a = arange(10) print a a = a.reshape(5,2) # 5 rows and 2 columns print a pycode-browser-1.02/Code/PythonBook/chap3/reshape.py000077500000000000000000000001261254404601000223430ustar00rootroot00000000000000#Example reshape.py from numpy import * a = arange(20) b = reshape(a, [4,5]) print b pycode-browser-1.02/Code/PythonBook/chap3/sine.py000077500000000000000000000001461254404601000216540ustar00rootroot00000000000000#Example sine.py import math x = 0.0 while x < math.pi * 4: print x , math.sin(x) x = x + 0.1pycode-browser-1.02/Code/PythonBook/chap3/vectorize.py000077500000000000000000000001531254404601000227260ustar00rootroot00000000000000from numpy import * def spf(x): return 3*x vspf = vectorize(spf) a = array([1,2,3,4]) print vspf(a) pycode-browser-1.02/Code/PythonBook/chap3/vfunc.py000077500000000000000000000000761254404601000220410ustar00rootroot00000000000000from numpy import * a = array([1,10,100,1000]) print log10(a) pycode-browser-1.02/Code/PythonBook/chap4/000077500000000000000000000000001254404601000203415ustar00rootroot00000000000000pycode-browser-1.02/Code/PythonBook/chap4/archi.py000077500000000000000000000001731254404601000220050ustar00rootroot00000000000000#Example archi.py from pylab import * a = 2 th= linspace(0, 10*pi,200) r = a*th polar(th,r) axis([0, 2*pi, 0, 70]) show() pycode-browser-1.02/Code/PythonBook/chap4/arcs.py000077500000000000000000000002511254404601000216440ustar00rootroot00000000000000#Example arcs.py from pylab import * a = 10.0 for a in range(5,21,5): th = linspace(0, pi * a/10, 200) x = a * cos(th) y = a * sin(th) plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/astro.py000077500000000000000000000001751254404601000220510ustar00rootroot00000000000000#Example astro.py from pylab import * a = 2 x = linspace(0,a,100) y = ( a**(2.0/3) - x**(2.0/3) )**(3.0/2) plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/astropar.py000077500000000000000000000002001254404601000225410ustar00rootroot00000000000000#Example astropar.py from pylab import * a = 2 t = linspace(-2*a,2*a,101) x = a * cos(t)**3 y = a * sin(t)**3 plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/circ.py000077500000000000000000000002451254404601000216370ustar00rootroot00000000000000#Example circ.py from pylab import * a = 10.0 x = linspace(-a, a , 200) yupper = sqrt(a**2 - x**2) ylower = -sqrt(a**2 - x**2) plot(x,yupper) plot(x,ylower) show() pycode-browser-1.02/Code/PythonBook/chap4/circpar.py000077500000000000000000000001771254404601000223460ustar00rootroot00000000000000#Example circpar.py from pylab import * a = 10.0 th = linspace(0, 2*pi, 200) x = a * cos(th) y = a * sin(th) plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/ellipse.py000077500000000000000000000002101254404601000223440ustar00rootroot00000000000000#Example ellipse.py from pylab import * a = 2 b = 3 t = linspace(0, 2 * pi, 100) x = a * sin(t) y = b * cos(t) plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/fermat.py000077500000000000000000000001761254404601000222000ustar00rootroot00000000000000#Example fermat.py from pylab import * a = 2 th= linspace(0, 10*pi,200) r = sqrt(a**2 * th) polar(th,r) polar(th, -r) show() pycode-browser-1.02/Code/PythonBook/chap4/fourier_sawtooth.py000077500000000000000000000003271254404601000243230ustar00rootroot00000000000000#Example fourier_sawtooth.py from pylab import * N = 100 # number of points x = linspace(-pi, pi, N) y = zeros(N) for n in range(1,10): term = (-1)**(n+1) * sin(n*x) / n y = y + term plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/fourier_square.py000077500000000000000000000003271254404601000237530ustar00rootroot00000000000000#Example fourier_square.py from pylab import * N = 100 # number of points x = linspace(0.0, 2 * pi, N) y = zeros(N) for n in range(5): term = sin((2*n+1)*x) / (2*n+1) y = y + term plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/imshow1.py000077500000000000000000000000721254404601000223040ustar00rootroot00000000000000from pylab import * m = random([50,50]) imshow(m) show() pycode-browser-1.02/Code/PythonBook/chap4/julia.py000077500000000000000000000021061254404601000220210ustar00rootroot00000000000000''' Region of a complex plane ranging from -1 to +1 in both real and imaginary axes is rpresented using a 2 dimensional matrix having X x Y elements.For X and Y equal to 200, the stepsize in the complex plane is 2.0/200 = 0.01. The nature of the pattern depends very much on the value of c. ''' from pylab import * X = 200 Y = 200 rlim = 1.0 ilim = 1.0 rscale = 2*rlim / X iscale = 2*ilim / Y MAXIT = 100 MAXABS = 2.0 c = 0.02 - 0.8j # The constant in equation z**2 + c m = zeros([X,Y],dtype=uint8) # A two dimensional array def numit(x,y): # number of iterations to diverge z = complex(x,y) for k in range(MAXIT): if abs(z) <= MAXABS: z = z**2 + c else: return k # diverged after k trials return MAXIT # did not diverge, for x in range(X): for y in range(Y): re = rscale * x - rlim # complex number represented im = iscale * y - ilim # by the (x,y) coordinate m[x][y] = numit(re,im) # get the color for (x,y) imshow(m) # Colored plot using the two dimensional matrix show() pycode-browser-1.02/Code/PythonBook/chap4/line3d.py000077500000000000000000000005051254404601000220740ustar00rootroot00000000000000from pylab import * from mpl_toolkits.mplot3d import Axes3D ax = Axes3D(figure()) phi = linspace(0, 2*pi, 400) x = cos(phi) y = sin(phi) z = 0 ax.plot(x, y, z, label = 'x') # circle z = sin(4*phi) # modulated in z plane ax.plot(x, y, z, label = 'x') ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_zlabel('Z') show() pycode-browser-1.02/Code/PythonBook/chap4/lissa.py000077500000000000000000000002611254404601000220300ustar00rootroot00000000000000#Example lissa.py from pylab import * a = 2 b = 3 t= linspace(0, 2*pi,100) x = a * sin(2*t) y = b * cos(t) plot(x,y) x = a * sin(3*t) y = b * cos(2*t) plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/mgrid1.py000077500000000000000000000001461254404601000221020ustar00rootroot00000000000000from numpy import * x = arange(0, 3, 1) y = arange(0, 3, 1) gx, gy = meshgrid(x, y) print gx print gy pycode-browser-1.02/Code/PythonBook/chap4/mgrid2.py000077500000000000000000000002231254404601000220770ustar00rootroot00000000000000 from pylab import * x = arange(-3*pi, 3*pi, 0.1) y = arange(-3*pi, 3*pi, 0.1) xx, yy = meshgrid(x, y) z = sin(xx) + sin(yy) imshow(z) show() pycode-browser-1.02/Code/PythonBook/chap4/npsin.py000077500000000000000000000002001254404601000220350ustar00rootroot00000000000000#Example npsin.py from pylab import * x = linspace(-pi, pi , 200) y = sin(x) y1 = sin(x*x) plot(x,y) plot(x,y1,'r') show() pycode-browser-1.02/Code/PythonBook/chap4/piechart.py000077500000000000000000000001711254404601000225140ustar00rootroot00000000000000from pylab import * labels = 'Frogs', 'Hogs', 'Dogs', 'Logs' fracs = [25, 25, 30, 20] pie(fracs, labels=labels) show() pycode-browser-1.02/Code/PythonBook/chap4/plot1.py000077500000000000000000000001101254404601000217450ustar00rootroot00000000000000#Example plot1.py from pylab import * data = [1,2,5] plot(data) show() pycode-browser-1.02/Code/PythonBook/chap4/plot2.py000077500000000000000000000001201254404601000217470ustar00rootroot00000000000000#Example plot2.py from pylab import * x = [1,2,5] y = [4,5,6] plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/plot3.py000077500000000000000000000002071254404601000217560ustar00rootroot00000000000000#Example plot3.py from pylab import * x = [1,2,5] y = [4,5,6] plot(x,y,'ro') xlabel('x-axis') ylabel('y-axis') axis([0,6,1,7]) show() pycode-browser-1.02/Code/PythonBook/chap4/plot4.py000077500000000000000000000002061254404601000217560ustar00rootroot00000000000000#Example plot4.py from pylab import * t = arange(0.0, 5.0, 0.2) plot(t, t**2,'x') # t^{2} plot(t, t**3,'ro') # t^{3} show() pycode-browser-1.02/Code/PythonBook/chap4/polar.py000077500000000000000000000001641254404601000220340ustar00rootroot00000000000000#Example polar.py from pylab import * th = linspace(0,2*pi,100) r = 5 * ones(100) # radius = 5 polar(th,r) show() pycode-browser-1.02/Code/PythonBook/chap4/rose.py000077500000000000000000000001321254404601000216620ustar00rootroot00000000000000from pylab import * k = 4 th = linspace(0, 12*pi,1000) r = cos(k*th) polar(th,r) show() pycode-browser-1.02/Code/PythonBook/chap4/series_sin.py000077500000000000000000000004631254404601000230640ustar00rootroot00000000000000#Example series_sin.py from pylab import * from scipy import factorial x = linspace(-pi, pi, 50) y = zeros(50) for n in range(5): term = (-1)**(n) * (x**(2*n+1)) / factorial(2*n+1) y = y + term #plot(x,term) plot(x, y, '+b') plot(x, sin(x),'r') # compare with the real one show() pycode-browser-1.02/Code/PythonBook/chap4/subplot1.py000077500000000000000000000002771254404601000224750ustar00rootroot00000000000000from pylab import * subplot(211) # the first subplot plot([1,2,3,4]) xlabel('first row') subplot(212) # the second subplot plot([4,2,3,1]) xlabel('second row') show() pycode-browser-1.02/Code/PythonBook/chap4/subplot2.py000077500000000000000000000004541254404601000224730ustar00rootroot00000000000000from pylab import * mark = ['x','o','^','+','>'] NR = 2 # number of rows NC = 3 # number of columns pn = 1 for row in range(NR): for col in range(NC): subplot(NR, NC, pn) a = rand(10) * pn plot(a, marker = mark[(pn+1)%5]) xlabel('plot %d X'%pn) ylabel('plot %d Y'%pn) pn = pn + 1 show() pycode-browser-1.02/Code/PythonBook/chap4/surf3d.py000077500000000000000000000002201254404601000221160ustar00rootroot00000000000000from numpy import * from enthought.mayavi.mlab import * x, y = mgrid[-10.:10.05:0.1, -10.:10.05:0.05] z = sin(x) + sin(y) surf(x, y, z) show() pycode-browser-1.02/Code/PythonBook/chap4/surface3d.py000077500000000000000000000004031254404601000225720ustar00rootroot00000000000000from pylab import * from mpl_toolkits.mplot3d import Axes3D ax = Axes3D(figure()) x = arange(-3*pi, 3*pi, 0.1) y = arange(-3*pi, 3*pi, 0.1) xx, yy = meshgrid(x, y) z = sin(xx) + sin(yy) ax.plot_surface(xx, yy, z, cmap=cm.jet, cstride=1) #imshow(z) show() pycode-browser-1.02/Code/PythonBook/chap4/wireframe.py000077500000000000000000000004771254404601000227070ustar00rootroot00000000000000from pylab import * from mpl_toolkits.mplot3d import Axes3D ax = Axes3D(figure()) phi = linspace(0, 2 * pi, 100) theta = linspace(0, pi, 100) x = 10 * outer(cos(phi), sin(theta)) y = 10 * outer(sin(phi), sin(theta)) z = 10 * outer(ones(size(phi)), cos(theta)) ax.plot_wireframe(x,y,z, rstride=2, cstride=2) show() pycode-browser-1.02/Code/PythonBook/chap4/xyplot.py000077500000000000000000000003301254404601000222510ustar00rootroot00000000000000from pylab import * x = [] y = [] f = open('xy.dat', 'r') while True: s = f.readline() if len(s) < 3: break ss = s.split() print ss x.append(ss[0]) y.append(ss[1]) plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap4/ylm20.py000077500000000000000000000004601254404601000216610ustar00rootroot00000000000000from numpy import * from enthought.mayavi import mlab polar = linspace(0,pi,100) azimuth = linspace(0, 2*pi,100) phi,th = meshgrid(polar, azimuth) r = 0.25 * sqrt(5.0/pi) * (3 * cos(phi) ** 2 - 1) #Ylm(0,2) x = r*sin(phi)*cos(th) y = r*cos(phi) z = r*sin(phi)*sin(th) mlab.mesh(x, y, z) mlab.show() pycode-browser-1.02/Code/PythonBook/chap5/000077500000000000000000000000001254404601000203425ustar00rootroot00000000000000pycode-browser-1.02/Code/PythonBook/chap5/environs.tex000077500000000000000000000010461254404601000227330ustar00rootroot00000000000000\documentclass{article} \begin{document} \begin{center} A numbered List (heading centered). \end{center} \begin{enumerate} \item dog \item cat \end{enumerate} \begin{flushleft} A bulleted list (heading justified left). \end{flushleft} \begin{itemize} \item dog \item cat \end{itemize} \begin{flushright} this part of the text is centered. \end{flushright} \begin{quote} Any text inside quote\\ environment will appe-\\ ar as typed.\\ \end{quote} \begin{verbatim} x = 1 while x <= 10: print x * 5 x = x + 1 \end{verbatim} \end{document} pycode-browser-1.02/Code/PythonBook/chap5/hello.tex000077500000000000000000000001151254404601000221670ustar00rootroot00000000000000\documentclass{article} \begin{document} Small is beautiful. \end{document} pycode-browser-1.02/Code/PythonBook/chap5/math1.tex000077500000000000000000000006151254404601000221030ustar00rootroot00000000000000\documentclass{article} \usepackage{amsmath} \begin{document} The equation $a^2 + b^2 = c^2$ is typeset as inline. The same can be done in a separate line using \begin{equation} a^2 + b^2 = c^2 \end{equation} We can also make the equation with a label refer to equation \eqref{mass} to know about the mass energy relation. \begin{equation} E = mc^2 \label{mass} \end{equation} \end{document} pycode-browser-1.02/Code/PythonBook/chap5/math_many.tex000077500000000000000000000010561254404601000230460ustar00rootroot00000000000000\documentclass{article} \usepackage{amsmath} \begin{document} $A\quad B\qquad C$\\ $ \alpha \beta \gamma \pi$\\ $A_n \quad A^m$\\ $a^b \quad a^{b^c}$\\ $\frac{3}{5}$\\ $n! = 1 \cdot 2 \cdots (n-1) \cdot n$\\ $0.\overline{3} = \underline{1/3}$\\ $\vec{a}$\\ $\sin x + \arctan y$\\ $\sqrt{x^2+y^2}$\\ $z=\sqrt[3]{x^{2} + \sqrt{y}}$\\ $A \neq B \quad A \approx C \quad $\\ $\Leftrightarrow\quad\Downarrow$\\ $\frac{\partial ^2f}{\partial x^2}$\\ $\prod_\epsilon$\\ $\Big((x+1)(x-1)\Big)^{2}$\\ $\int_a^b f(x) dx$\\ $\pm \div \times \cup \ast $ \end{document} pycode-browser-1.02/Code/PythonBook/chap5/qpaper.tex000077500000000000000000000015421254404601000223610ustar00rootroot00000000000000\documentclass{article} \usepackage{amsmath} \begin{document} \begin{center} \large{\textbf{Sample Question Paper\\for\\Mathematics using Python}} \end{center} \begin{tabular}{p{8cm}r} \textbf{Duration:3 Hrs} & \textbf{30 weightage} \end{tabular}\\ \section{Answer all Questions. $4\times 1\frac{1}{2}$} \begin{enumerate} \item What are the main document classes supported by LaTeX. \item Typeset $\sin^{2}x+\cos^{2}x=1$ using LaTeX. \item Plot a circle using the polar() function. \item Write code to print all perfect cubes upto 2000. \end{enumerate} \section{Answer any two Questions. $3\times 5$} \begin{enumerate} \item Set a sample question paper using LaTeX. \item Write a Python function to calculate the GCD of two numbers \item Write a program with a Canvas and a circle drawn on it. \end{enumerate} \begin{center}\text{End}\end{center} \end{document} pycode-browser-1.02/Code/PythonBook/chap5/sections.tex000077500000000000000000000004771254404601000227260ustar00rootroot00000000000000\documentclass{article} \begin{document} \tableofcontents \section{Animals} This document defines sections. \subsection{Domestic} This document also defines subsections. \subsubsection{cats and dogs} Cats and dogs are Domestic animals. \section*{Appendix A} The section with a * is not given a number. \end{document} pycode-browser-1.02/Code/PythonBook/chap5/texts.tex000077500000000000000000000003701254404601000222360ustar00rootroot00000000000000\documentclass{article} \begin{document} This is normal text. \newline \textbf{This is bold face text. } \textit{This is italic text. }\\ \tiny{This is tiny text. } \large{This is large text. } \underline{This is underlined text.} \end{document} pycode-browser-1.02/Code/PythonBook/chap6/000077500000000000000000000000001254404601000203435ustar00rootroot00000000000000pycode-browser-1.02/Code/PythonBook/chap6/bisection.py000077500000000000000000000011311254404601000226730ustar00rootroot00000000000000import math def func(x): return x**3 - 10.0* x*x + 5 def bisect(f, x1, x2, epsilon=1.0e-9): f1 = f(x1) f2 = f(x2) if f1*f2 > 0.0: print 'x1 ans x2 are on the same side of x-axis' return None n = math.ceil(math.log(abs(x2 - x1)/epsilon)/math.log(2.0)) n = int(n) for i in range(n): x3 = 0.5 * (x1 + x2) f3 = f(x3) if f3 == 0.0: return x3 if f2*f3 < 0.0: x1 = x3 f1 = f3 else: x2 =x3 f2 = f3 return (x1 + x2)/2.0 print bisect(func, 0.70, 0.8, 1.0e-4) print bisect(func, 0.70, 0.8, 1.0e-9) pycode-browser-1.02/Code/PythonBook/chap6/compareEuRK4.py000077500000000000000000000013751254404601000231670ustar00rootroot00000000000000''' Cosine function is integrated by Euler and RK4 methods. Results at each step is compared with sine function. ''' from pylab import * def rk4(x, y, fx, h = 0.1): # x, y , derivative, stepsize k1 = h * fx(x) k2 = h * fx(x + h/2.0) k3 = h * fx(x + h/2.0) k4 = h * fx(x + h) return y + ( k1/6 + k2/3 + k3/3 + k4/6 ) h = 0.1 # stepsize x = 0.0 # initail values ye = 0.0 # for Euler yr = 0.0 # for RK4 ax = [] # Lists to store calculated values euerr = [] rkerr = [] while x < 2*pi: ye = ye + h * math.cos(x) # Euler method yr = rk4(x, yr, cos, h) # Runge-Kutta method x = x + h ax.append(x) euerr.append(ye - sin(x)) rkerr.append(yr - sin(x)) plot(ax,euerr,'+') plot(ax, rkerr,'ro') show() pycode-browser-1.02/Code/PythonBook/chap6/diff.py000077500000000000000000000005241254404601000216310ustar00rootroot00000000000000from math import * def f1(x): return x**2 def f2(x): return x**4 def f3(x): return x**10 def deriv(func, x, dx=0.1): df = func(x+dx/2)-func(x-dx/2) return df/dx print deriv(f1, 1.0), deriv(f1, 1.0, 0.01) print deriv(f2, 1.0), deriv(f2, 1.0, 0.01) print deriv(f3, 1.0), deriv(f3, 1.0, 0.01) pycode-browser-1.02/Code/PythonBook/chap6/euler.py000077500000000000000000000004531254404601000220360ustar00rootroot00000000000000#integrate cosine using Euler method. The result should be sine from pylab import * h = 0.01 # stepsize x = 0.0 # initail values y = 0.0 ax = [] ay = [] while x < 2*pi: y = y + h * math.cos(x) # Euler method x = x + h ax.append(x) ay.append(y) plot(ax,ay) show() pycode-browser-1.02/Code/PythonBook/chap6/lsfit.py000077500000000000000000000003731254404601000220440ustar00rootroot00000000000000from pylab import * NP = 50 r = 2*ranf([NP]) - 0.5 x = linspace(0,10,NP) data = 3 * x + 2 + r xbar = mean(x) ybar = mean(data) b = sum(data*(x-xbar)) / sum(x*(x-xbar)) a = ybar - xbar * b print a,b y = a + b * x plot(x,y) plot(x,data,'ob') show() pycode-browser-1.02/Code/PythonBook/chap6/newinterpol.py000077500000000000000000000010051254404601000232620ustar00rootroot00000000000000from pylab import * def eval(a,xpoints,x): n = len(xpoints) - 1 p = a[n] for k in range(1,n+1): p = a[n-k] + (x -xpoints[n-k]) * p return p def coef(x,y): a = copy(y) m = len(x) for k in range(1,m): a[k:m] = (a[k:m] - a[k-1])/(x[k:m]-x[k-1]) return a x = array([0,1,2,3]) y = array([0,3,14,39]) coef = coef(x,y) NP = 20 newx = linspace(0,3, NP) # New x-values newy = zeros(NP) for i in range(NP): newy[i] = eval(coef, x, newx[i]) plot(newx, newy,'-x') plot(x, y,'ro') show() pycode-browser-1.02/Code/PythonBook/chap6/newpoly.py000077500000000000000000000003631254404601000224170ustar00rootroot00000000000000from copy import copy def coef(x,y): a = copy(y) m = len(x) for k in range(1,m): tmp = copy(a) for i in range(k,m): tmp[i] = (a[i] - a[i-1])/(x[i]-x[i-k]) a = copy(tmp) return a x = [0,1,2,3] y = [0,3,14,39] print coef(x,y) pycode-browser-1.02/Code/PythonBook/chap6/newpoly2.py000077500000000000000000000003131254404601000224740ustar00rootroot00000000000000from numpy import * def coef(x,y): a = copy(y) m = len(x) for k in range(1,m): a[k:m] = (a[k:m] - a[k-1])/(x[k:m]-x[k-1]) return a x = array([0,1,2,3]) y = array([0,3,14,39]) print coef(x,y) pycode-browser-1.02/Code/PythonBook/chap6/newraph.py000077500000000000000000000003751254404601000223710ustar00rootroot00000000000000from pylab import * def f(x): return 2.0 * x**2 - 3*x - 5 def df(x): return 4.0 * x - 3 def nr(x, tol = 1.0e-9): for i in range(30): dx = -f(x)/df(x) #print x x = x + dx if abs(dx) < tol: return x print nr(4) print nr(1) print nr(-4) pycode-browser-1.02/Code/PythonBook/chap6/newraph_plot.py000077500000000000000000000004661254404601000234300ustar00rootroot00000000000000#Newton Raphson method from pylab import * def f(x): return 2.0 * x**2 - 3*x - 5 def df(x): return 4.0 * x - 3 vf = vectorize(f) x = linspace(-2, 5, 100) y = vf(x) x1 = 4 tg1 = df(x1)*(x-x1) + f(x1) x1 = -3 tg2 = df(x1)*(x-x1) + f(x1) grid(True) plot(x,y) plot(x,tg1) plot(x,tg2) ylim([-20,40]) show() pycode-browser-1.02/Code/PythonBook/chap6/poly.py000077500000000000000000000002761254404601000217100ustar00rootroot00000000000000#Example poly.py from pylab import * a = poly1d([3,4,5]) b = poly1d([6,7]) c = a * b + 5 d = c/a print a print a(0.5) print b print a * b print a.deriv() print a.integ() print d[0], d[1] pycode-browser-1.02/Code/PythonBook/chap6/polyplot.py000077500000000000000000000003011254404601000225740ustar00rootroot00000000000000#Example polyplot.py from pylab import * x = linspace(-pi, pi, 100) a = poly1d([-1.0/5040,0,1.0/120,0,-1.0/6,0,1,0]) da = a.deriv() y = a(x) y1 = da(x) plot(x,y) plot(x,y1) grid(True) show() pycode-browser-1.02/Code/PythonBook/chap6/rk4.py000077500000000000000000000010751254404601000214230ustar00rootroot00000000000000''' Solving initial value problem using 4th order Runge-Kutta method. Sine function is calculated by integrating its derivative, cosine. ''' from pylab import * def rk4(x, y, fx, h = 0.01): # x, y , derivative, stepsize k1 = h * fx(x) k2 = h * fx(x + h/2.0) k3 = h * fx(x + h/2.0) k4 = h * fx(x + h) return y + ( k1/6 + k2/3 + k3/3 + k4/6 ) h = 0.01 # stepsize x = 0.0 # initail values y = 0.0 ax = [x] ay = [y] while x < math.pi: y = rk4(x,y,math.cos) # Runge-Kutta method x = x + h ax.append(x) ay.append(y) plot(ax,ay) show() pycode-browser-1.02/Code/PythonBook/chap6/rk4_proper.py000077500000000000000000000011701254404601000230060ustar00rootroot00000000000000''' Solving initial value problem using 4th order Runge-Kutta method. derivative depends on the value of the function. f(x,y) = 1 + y**2 ''' from pylab import * def f1(x,y): return 1 + y**2 def f2(x,y): return (y-x)/(y+x) def rk4(x, y, fxy, h): # x, y , f(x,y) k1 = h * fxy(x, y) k2 = h * fxy(x + h/2.0, y+k1/2) k3 = h * fxy(x + h/2.0, y+k2/2) k4 = h * fxy(x + h, y+k3) ny = y + ( k1/6 + k2/3 + k3/3 + k4/6 ) #print x,y,k1,k2,k3,k4, ny return ny h = 0.2 # stepsize x = 0.0 # initail values y = 0.0 print rk4(x,y,f1,h) h = 1 # stepsize x = 0.0 # initail values y = 1.0 print rk4(x,y,f2,h) pycode-browser-1.02/Code/PythonBook/chap6/rootsearch.py000077500000000000000000000005051254404601000230710ustar00rootroot00000000000000import math def func(x): return x**3 - 10.0* x*x + 5 def rootsearch(f,a,b,dx): x = a while True: f1 = f(x) f2 = f(x+dx) if f1*f2 < 0: return x, x + dx x = x + dx if x >= b: print 'Failed to find root' return x,y = rootsearch(func, 0.0,1.0,.1) print x,y x,y = rootsearch(math.cos, 0.0, 4,.1) print x,y pycode-browser-1.02/Code/PythonBook/chap6/series_sc.py000077500000000000000000000006121254404601000226760ustar00rootroot00000000000000from pylab import * def factorial(n): # a recursive function if n == 0: return 1 else: return n * factorial(n-1) NP = 100 x = linspace(-pi, pi, NP) sinx = zeros(NP) cosx = zeros(NP) for n in range(10): sinx += (-1)**(n) * (x**(2*n+1)) / factorial(2*n+1) cosx += (-1)**(n) * (x**(2*n)) / factorial(2*n) plot(x, sinx) plot(x, cosx, 'r') show() pycode-browser-1.02/Code/PythonBook/chap6/solve_eqn.py000077500000000000000000000001551254404601000227140ustar00rootroot00000000000000from numpy import * b = array([0,9,0]) A = array([ [4,1,-2], [2,-3,3],[-6,-2,1]]) print dot(linalg.inv(A),b) pycode-browser-1.02/Code/PythonBook/chap6/tan.py000077500000000000000000000001131254404601000214750ustar00rootroot00000000000000from pylab import * x = linspace(0, 2*pi,400) y = tan(x) plot(x,y) show() pycode-browser-1.02/Code/PythonBook/chap6/taylor.py000077500000000000000000000004301254404601000222270ustar00rootroot00000000000000from numpy import * p = poly1d([1,1,1,0]) dp = p.deriv() dp2 = dp.deriv() dp3 = dp2.deriv() a = 0 # The known point x = 0 while x < .5: tay = p(a) + (x-a)* dp(a) + \ (x-a)**2 * dp2(a) / 2 + (x-a)**3 * dp3(a)/6 print '%5.1f %8.5f\t%8.5f'%(x, p(x), tay) x = x + .1 pycode-browser-1.02/Code/PythonBook/chap6/trapez.py000077500000000000000000000004721254404601000222300ustar00rootroot00000000000000from math import * def sqr(a): return sqrt(1.0 - a**2) def trapez(f, a, b, n): h = (b-a) / n sum = f(a) for i in range (1,n): sum = sum + 2 * f(a + h * i) sum = sum + f(b) return 0.5 * h * sum print 4*trapez(sqr,0.,1.,100) print 4*trapez(sqr,0.,1.,1000) print 4*trapez(sqr,0,1,100) # Why the error ? pycode-browser-1.02/Code/PythonBook/chap6/trapez_figure.py000077500000000000000000000005351254404601000235710ustar00rootroot00000000000000#Example polyplot.py from pylab import * x = linspace(0.5,2.5, 100) y = 1 + sin(x) plot(x,y,linewidth=2) a = [1,1] b = [0,1+sin(1.)] plot(a,b,'b-',linewidth=2) a = [2,2] b = [0,1+sin(2.0)] plot(a,b,'b-',linewidth=2) xc = 1+.05 while xc < 2: a = [xc,xc] b = [0,1+sin(xc)] plot(a,b,'b-',linewidth=1) xc= xc + .1 xlim(0.,3) ylim(0.,3) show() pycode-browser-1.02/Code/PythonBook/chap6/vdiff.py000077500000000000000000000004131254404601000220140ustar00rootroot00000000000000#Example vdiff.py from pylab import * def f(x): return sin(x) def deriv(x,dx=0.005): df = f(x+dx/2)-f(x-dx/2) return df/dx vecderiv = vectorize(deriv) x = linspace(-2*pi, 2*pi, 200) y = vecderiv(x,1.0) plot(x,y,'+') plot(x,cos(x)) show() pycode-browser-1.02/LICENSE.txt000066400000000000000000001045131254404601000162230ustar00rootroot00000000000000 GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. 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However, nothing other than this License grants you permission to propagate or modify any covered work. These actions infringe copyright if you do not accept this License. Therefore, by modifying or propagating a covered work, you indicate your acceptance of this License to do so. 10. Automatic Licensing of Downstream Recipients. Each time you convey a covered work, the recipient automatically receives a license from the original licensors, to run, modify and propagate that work, subject to this License. You are not responsible for enforcing compliance by third parties with this License. An "entity transaction" is a transaction transferring control of an organization, or substantially all assets of one, or subdividing an organization, or merging organizations. If propagation of a covered work results from an entity transaction, each party to that transaction who receives a copy of the work also receives whatever licenses to the work the party's predecessor in interest had or could give under the previous paragraph, plus a right to possession of the Corresponding Source of the work from the predecessor in interest, if the predecessor has it or can get it with reasonable efforts. You may not impose any further restrictions on the exercise of the rights granted or affirmed under this License. For example, you may not impose a license fee, royalty, or other charge for exercise of rights granted under this License, and you may not initiate litigation (including a cross-claim or counterclaim in a lawsuit) alleging that any patent claim is infringed by making, using, selling, offering for sale, or importing the Program or any portion of it. 11. Patents. A "contributor" is a copyright holder who authorizes use under this License of the Program or a work on which the Program is based. The work thus licensed is called the contributor's "contributor version". A contributor's "essential patent claims" are all patent claims owned or controlled by the contributor, whether already acquired or hereafter acquired, that would be infringed by some manner, permitted by this License, of making, using, or selling its contributor version, but do not include claims that would be infringed only as a consequence of further modification of the contributor version. For purposes of this definition, "control" includes the right to grant patent sublicenses in a manner consistent with the requirements of this License. Each contributor grants you a non-exclusive, worldwide, royalty-free patent license under the contributor's essential patent claims, to make, use, sell, offer for sale, import and otherwise run, modify and propagate the contents of its contributor version. In the following three paragraphs, a "patent license" is any express agreement or commitment, however denominated, not to enforce a patent (such as an express permission to practice a patent or covenant not to sue for patent infringement). To "grant" such a patent license to a party means to make such an agreement or commitment not to enforce a patent against the party. 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You may not convey a covered work if you are a party to an arrangement with a third party that is in the business of distributing software, under which you make payment to the third party based on the extent of your activity of conveying the work, and under which the third party grants, to any of the parties who would receive the covered work from you, a discriminatory patent license (a) in connection with copies of the covered work conveyed by you (or copies made from those copies), or (b) primarily for and in connection with specific products or compilations that contain the covered work, unless you entered into that arrangement, or that patent license was granted, prior to 28 March 2007. Nothing in this License shall be construed as excluding or limiting any implied license or other defenses to infringement that may otherwise be available to you under applicable patent law. 12. No Surrender of Others' Freedom. If conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot convey a covered work so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not convey it at all. For example, if you agree to terms that obligate you to collect a royalty for further conveying from those to whom you convey the Program, the only way you could satisfy both those terms and this License would be to refrain entirely from conveying the Program. 13. Use with the GNU Affero General Public License. Notwithstanding any other provision of this License, you have permission to link or combine any covered work with a work licensed under version 3 of the GNU Affero General Public License into a single combined work, and to convey the resulting work. The terms of this License will continue to apply to the part which is the covered work, but the special requirements of the GNU Affero General Public License, section 13, concerning interaction through a network will apply to the combination as such. 14. Revised Versions of this License. The Free Software Foundation may publish revised and/or new versions of the GNU General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Program specifies that a certain numbered version of the GNU General Public License "or any later version" applies to it, you have the option of following the terms and conditions either of that numbered version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of the GNU General Public License, you may choose any version ever published by the Free Software Foundation. If the Program specifies that a proxy can decide which future versions of the GNU General Public License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Program. Later license versions may give you additional or different permissions. However, no additional obligations are imposed on any author or copyright holder as a result of your choosing to follow a later version. 15. Disclaimer of Warranty. THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 16. Limitation of Liability. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. 17. Interpretation of Sections 15 and 16. If the disclaimer of warranty and limitation of liability provided above cannot be given local legal effect according to their terms, reviewing courts shall apply local law that most closely approximates an absolute waiver of all civil liability in connection with the Program, unless a warranty or assumption of liability accompanies a copy of the Program in return for a fee. END OF TERMS AND CONDITIONS How to Apply These Terms to Your New Programs If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms. To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively state the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found. Copyright (C) This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . Also add information on how to contact you by electronic and paper mail. If the program does terminal interaction, make it output a short notice like this when it starts in an interactive mode: Copyright (C) This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details. The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, your program's commands might be different; for a GUI interface, you would use an "about box". You should also get your employer (if you work as a programmer) or school, if any, to sign a "copyright disclaimer" for the program, if necessary. For more information on this, and how to apply and follow the GNU GPL, see . The GNU General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. But first, please read . pycode-browser-1.02/Makefile000066400000000000000000000013171254404601000160360ustar00rootroot00000000000000DESTDIR= SHAREDIR=/usr/share/pycode-browser BINDIR=$(DESTDIR)/usr/bin APPDIR=$(DESTDIR)/usr/share/applications all: make -C pybooksrc all DESTDIR=$(DESTDIR) clean: rm -f *~ make -C pybooksrc clean install: mkdir -p $(DESTDIR)$(SHAREDIR) cp -a Code gui pycode-browser.py $(DESTDIR)$(SHAREDIR) cp pybooksrc/mapy.pdf $(DESTDIR)$(SHAREDIR)/ find $(DESTDIR)$(SHAREDIR) -type f -exec chmod 466 {} \; mkdir $(BINDIR) echo "#! /bin/sh" > $(BINDIR)/pycode-browser echo "python $(SHAREDIR)/pycode-browser.py" >> $(BINDIR)/pycode-browser echo "#! /bin/sh" > $(BINDIR)/pycode-browser-book echo "evince $(SHAREDIR)/mapy.pdf" >> $(BINDIR)/pycode-browser-book mkdir -p $(APPDIR) install -m 644 *.desktop $(APPDIR) pycode-browser-1.02/README000066400000000000000000000010771254404601000152610ustar00rootroot00000000000000PyCodeBrowser: You can put any number of python files in this directory and the software will list and execute the files. Note: Most of the example code on the Code directory were written by Dr. Ajith Kumar for phoenix project. Rest are from the example code for various projects. If you found any of these sample programs are non-distributable please inform me, I will remove it from the repository. Version 0.93 Authors: Vibeesh P Vimal Joseph Python code in the Code directory: Dr. Ajith Kumar pycode-browser-1.02/gui/000077500000000000000000000000001254404601000151605ustar00rootroot00000000000000pycode-browser-1.02/gui/gui.glade000077500000000000000000000257231254404601000167560ustar00rootroot00000000000000 800 600 True Python Code Browser center True vertical True True _File True True gtk-execute True False True True gtk-save-as True False True True True gtk-quit True True True True _Help True True gtk-about True True True False 0 True both True False Execute True gtk-execute False True True False Save As True gtk-save-as False True True Quit True gtk-quit False True False 1 True True True 250 True True in 150 407 True True False True True vertical True True automatic automatic 0 True True 0 2 True True 300 True True never automatic True Terminal label_item 3 True 2 False 4 pycode-browser-1.02/gui/gui.ui000066400000000000000000000231641254404601000163110ustar00rootroot00000000000000 menuitem1 _File gtk-execute btnExecute gtk-save-as btnSaveas gtk-quit imagemenuitem5 menuitem4 _Help gtk-about imagemenuitem10 800 600 True Python Code Browser center True vertical True False 0 True both True False Execute True gtk-execute False True True False Save As True gtk-save-as False True True Quit True gtk-quit False True False 1 True True True 250 True True in 150 407 True True False True True vertical True True automatic automatic 0 True True 0 2 True True 300 True True never automatic True Terminal 3 True 2 False 4 pycode-browser-1.02/pybooksrc/000077500000000000000000000000001254404601000164075ustar00rootroot00000000000000pycode-browser-1.02/pybooksrc/GNU-fdl.txt000066400000000000000000000546621254404601000203610ustar00rootroot00000000000000 GNU Free Documentation License Version 1.3, 3 November 2008 Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference. 1. APPLICABILITY AND DEFINITIONS This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you". 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To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles. You may add a section Entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various parties--for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one. The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version. 5. COMBINING DOCUMENTS You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. 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TERMINATION You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License. However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation. Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it. 10. FUTURE REVISIONS OF THIS LICENSE The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document. 11. RELICENSING "Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. 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The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing. ADDENDUM: How to use this License for your documents To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: Copyright (c) YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this: with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software. pycode-browser-1.02/pybooksrc/LICENSE.TXT000066400000000000000000000003011254404601000200640ustar00rootroot00000000000000This Document is distributed under the GNU Free Documentation License. Author : Ajith Kumar B.P., Inter University Accelerator Centre, New Delhi 110067. (bpajith@gmail.com, ajith@iuac.res.in) pycode-browser-1.02/pybooksrc/Makefile000066400000000000000000000005631254404601000200530ustar00rootroot00000000000000.SUFFIXES= .lyx .tex .pdf DESTDIR= all: mapy.pdf clean: rm -f *~ *.tex *.aux *.log *.pdf *.toc pics/*.eps install: %.tex: %.lyx tmpHome=$$(mktemp -d); \ HOME=$${tmpHome} lyx --export latex $<; \ rm -rf $${tmpHome} %.pdf: %.tex pdflatex $< logfile=$$(echo $< | sed 's/tex/log/'); \ while (grep Warning $${logfile}| grep -iq run); do \ pdflatex $<; \ done pycode-browser-1.02/pybooksrc/mapy.lyx000066400000000000000000012066101254404601000201210ustar00rootroot00000000000000#LyX 1.5.5 created this file. For more info see http://www.lyx.org/ \lyxformat 276 \begin_document \begin_header \textclass book \language american \inputencoding auto \font_roman default \font_sans default \font_typewriter default \font_default_family default \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \paperfontsize default \spacing single \papersize custom \use_geometry true \use_amsmath 1 \use_esint 0 \cite_engine basic \use_bibtopic false \paperorientation portrait \paperwidth 21cm \paperheight 24cm \leftmargin 3cm \topmargin 1.5cm \rightmargin 3cm \bottommargin 1.5cm \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \author "" \author "" \end_header \begin_body \begin_layout Standard \lang english \begin_inset ERT status open \begin_layout Standard \backslash thispagestyle{empty} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset VSpace 0.5in* \end_inset \end_layout \begin_layout Standard \align center \size huge Python for Education \end_layout \begin_layout Standard \align center \begin_inset Graphics filename pics/ylm20.png width 8cm \end_inset \end_layout \begin_layout Standard \align center \shape italic \size large Learning Maths & Science using Python \end_layout \begin_layout Standard \align center \shape italic \size large and \end_layout \begin_layout Standard \align center \shape italic \size large writing them in LaTeX \end_layout \begin_layout Standard \begin_inset VSpace 0.5in \end_inset \end_layout \begin_layout Standard \align center \size large Ajith Kumar B.P. \end_layout \begin_layout Standard \align center \size large Inter University Accelerator Centre \end_layout \begin_layout Standard \align center \size large New Delhi 110067 \end_layout \begin_layout Standard \align center \size large www.iuac.res.in \end_layout \begin_layout Standard \align center \begin_inset VSpace 0.3in \end_inset \end_layout \begin_layout Standard \align center June 2010 \end_layout \begin_layout Standard \newpage \end_layout \begin_layout Standard \align center Preface \end_layout \begin_layout Standard \begin_inset Quotes eld \end_inset Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show \begin_inset Quotes erd \end_inset , wrote Bertrand Russell about the beauty of mathematics. All of us may not reach such higher planes, probably reserved for Russels and Ramanujans, but we also have beautiful curves and nice geometrical figures with intricate symmetries, like fractals, generated by seemingly dull equations. This book attempts to explore it using a simple tool, the Python programming language. \end_layout \begin_layout Standard I started using Python for the Phoenix project (www.iuac.res.in). Phoenix was driven in to Python by Pramode CE (pramode.net) and I followed. Writing this document was triggered by some of my friends who are teaching mathematics at Calicut University. \end_layout \begin_layout Standard In the first chapter, a general introduction about computers and high level programming languages is given. Basics of Python language, Python modules for array and matrix manipulation, 2D and 3D data visualization, type-setting mathematical equations using latex and numerical methods in Python are covered in the subsequent chapters. Example programs are given for every topic discussed. This document is meant for those who want to try out these examples and modify them for better understanding. Huge amount of material is already available online on the topics covered, and the references to many resources on the Internet are given for the benefit of the serious reader. \end_layout \begin_layout Standard This book comes with a live CD, containing a modified version of Ubuntu GNU/Linux operating system. You can boot any PC from this CD and practice Python. Click on the 'Learn by Coding' icon on the desktop to browse through a collection of Python programs, run any of them with a single click. You can practice Python very easily by modifying and running these example programs. \end_layout \begin_layout Standard This document is prepared using LyX, a LaTeX front-end. It is distributed under the GNU Free Documentation License (www.gnu.org). Feel free to make verbatim copies of this document and distribute through any media. For the LyX source files please contact the author. \end_layout \begin_layout Standard Ajith Kumar \end_layout \begin_layout Standard IUAC , New Delhi \end_layout \begin_layout Standard ajith at iuac.res.in \end_layout \begin_layout Standard \newpage \end_layout \begin_layout Standard \begin_inset LatexCommand tableofcontents \end_inset \end_layout \begin_layout Chapter Introduction \end_layout \begin_layout Standard Primary objective of this book is to explore the possibilities of using Python language as a tool for learning mathematics and science. The reader is not assumed to be familiar with computer programming. Ability to think logically is enough. Before getting into Python programming, we will briefly explain some basic concepts and tools required. \end_layout \begin_layout Standard Computer is essentially an electronic device like a radio or a television. What makes it different from a radio or a TV is its ability to perform different kinds of tasks using the same electronic and mechanical components. This is achieved by making the electronic circuits flexible enough to work according to a set of instructions. The electronic and mechanical parts of a computer are called the Hardware and the set of instructions is called Software (or computer program). Just by changing the Software, computer can perform vastly different kind of tasks. The instructions are stored in binary format using electronic switches. \end_layout \begin_layout Section Hardware Components \end_layout \begin_layout Standard Central Processing Unit (CPU), Memory and Input/Output units are the main hardware components of a computer. CPU \begin_inset Foot status collapsed \begin_layout Standard The cabinet that encloses most of the hardware is called CPU by some, mainly the computer vendors. They are not referring to the actual CPU chip. \end_layout \end_inset can be called the brain of the computer. It contains a Control Unit and an Arithmetic and Logic Unit, ALU. The control unit brings the instructions stored in the main memory one by one and acts according to it. It also controls the movement of data between memory and input/output units. The ALU can perform arithmetic operations like addition, multiplication and logical operations like comparing two numbers. \end_layout \begin_layout Standard Memory stores the instructions and data, that is processed by the CPU. All types of information are stored as binary numbers. The smallest unit of memory is called a binary digit or Bit. It can have a value of zero or one. A group of eight bits are called a Byte. A computer has Main and Secondary types of memory. Before processing, data and instructions are moved into the main memory. Main memory is organized into words of one byte size. CPU can select any memory location by using it's address. Main memory is made of semiconductor switches and is very fast. There are two types of Main Memory. Read Only Memory and Read/Write Memory. Read/Write Memory is also called Random Access Memory. All computers contains some programs in ROM which start running when you switch on the machine. Data and programs to be stored for future use are saved to Secondary memory, mainly devices like Hard disks, floppy disks, CDROM or magnetic tapes. \end_layout \begin_layout Standard The Input devices are for feeding the input data into the computer. Keyboard is the most common input device. Mouse, scanner etc. are other input devices. The processed data is displayed or printed using the output devices. The monitor screen and printer are the most common output devices. \end_layout \begin_layout Section Software components \end_layout \begin_layout Standard An ordinary user expects an easy and comfortable interaction with a computer, and most of them are least inclined to learn even the basic concepts. To use modern computers for common applications like browsing and word processing, all you need to do is to click on some icons and type on the keyboard. However, to write your own computer programs, you need to learn some basic concepts, like the operating system, editors, compilers, different types of user interfaces etc. This section describes the basics from that point of view. \end_layout \begin_layout Subsection The Operating System \end_layout \begin_layout Standard Operating system (OS) is the software that interacts with the user and makes the hardware resources available to the user. It starts running when you switch on the computer and remains in control. On user request, operating system loads other application programs from disk to the main memory and executes them. OS also provides a file system, a facility to store information on devices like floppy disk and hard disk. In fact the OS is responsible for managing all the hardware resources. \end_layout \begin_layout Standard GNU/Linux and MS Windows are two popular operating systems. Based on certain features, operating systems can be classified as: \end_layout \begin_layout Itemize Single user, single process systems like MS DOS. Only one process can run at a time. Such operating systems do not have much control over the application programs. \end_layout \begin_layout Itemize Multi-tasking systems like MS Windows, where more than one processe can run at a time. \end_layout \begin_layout Itemize Multi-user, multi-tasking systems like GNU/Linux, Unix etc. More than one person can use the computer at the same time. \end_layout \begin_layout Itemize Real-time systems, mostly used in control applications, where the response time to any external input is maintained under specified limits. \end_layout \begin_layout Subsection The User Interface \end_layout \begin_layout Standard Interacting with a computer involves, starting various application programs and managing them on the computer screen. The software that manages these actions is called the user interface. The two most common forms of user interface have historically been the Command-line Interface, where computer commands are typed out line-by-line, and the Graphical User Interface (GUI), where a visual environment (consisting of windows, menus, buttons, icons, etc.) is present. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/terminal.png width 12cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard A GNU/Linux Terminal \begin_inset LatexCommand label name "fig:The-command-terminal" \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Subsubsection The Command Terminal \end_layout \begin_layout Standard To run any particular program, we need to request the operating system to do so. Under a Graphical User Interface, we do this by choosing the desired applicatio n from a menu. It is possible only because someone has added it to the menu earlier. When you start writing your own programs, obviously they will not appear in any menu. Another way to request the operating system to execute a program is to enter the name of the program (more precisely, the name of the file containing it) at the Command Terminal. On an Ubuntu GNU/Linux system, you can start a Terminal from the menu names Applications->Accessories->Terminal. Figure \begin_inset LatexCommand ref reference "fig:The-command-terminal" \end_inset shows a Terminal displaying the list of files in a directory (output of the command 'ls -l' , the -l option is for long listing). \end_layout \begin_layout Standard The command processor offers a host of features to make the interaction more comfortable. It keeps track of the history of commands and we can recall previous commands, modify and reuse them using the cursor keys. There is also a completion feature implemented using the Tab key that reduces typing. Use the tab key to complete command and filenames. To run \shape italic hello.py \shape default from our test directory, type \shape italic python h \shape default and then press the tab key to complete it. If there are more than one file starting with 'h', you need to type more characters until the ambiguity is removed. Always use the up-cursor key to recall the previous commands and re-issue it. \end_layout \begin_layout Standard The commands given at the terminal are processed by a program called the \shape italic shell \shape default . (The version now popular under GNU/Linux is called bash, the Bourne again shell). Some of the GNU/Linux commands are listed below. \end_layout \begin_layout Itemize top : Shows the CPU and memory usage of all the processes started. \end_layout \begin_layout Itemize cp filename filename : copies a file to another. \end_layout \begin_layout Itemize mv : moves files from one folder to another, or rename a file. \end_layout \begin_layout Itemize rm : deletes files or directories. \end_layout \begin_layout Itemize man : display manual pages for a program. For example 'man bash' will give help on the bash shell. Press 'q' to come out of the help screen. \end_layout \begin_layout Itemize info : A menu driven information system on various topics. \end_layout \begin_layout Standard See the manual pages of 'mv', cp, 'rm' etc. to know more about them. Most of these commands are application programs, stored inside the folders /bin or /sbin, that the shell starts for you and displays their output inside the terminal window. \end_layout \begin_layout Subsection The File-system \end_layout \begin_layout Standard Before the advent of computers, people used to keep documents in files and folders. The designers of the Operating System have implemented the electronic counterpa rts of the same. The storage space is made to appear as files arranged inside folders (directory is another term for folder). A simplified schematic of the GNU/Linux file system is shown in figure \begin_inset LatexCommand ref reference "fig:The-GNU/Linux-file" \end_inset . The outermost directory is called 'root' directory and represented using the forward slash character. Inside that we have folders named bin, usr, home, tmp etc., containing different type of files. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/linux-tree.png width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard The GNU/Linux file system tree \begin_inset LatexCommand label name "fig:The-GNU/Linux-file" \end_inset . \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Ownership & permissions \end_layout \begin_layout Standard On a multi-user operating system, application programs and document files must be protected against any misuse. This is achieved by defining a scheme of ownerships and permissions. Each and every file on the system will be owned by a specific user. The read, write and execute permissions can be assigned to them, to control the usage. The concept of \shape italic group \shape default is introduced to share files between a selected group of users. \end_layout \begin_layout Standard There is one special user named \shape italic root \shape default (also called the system administrator or the super user) , who has permission to use all the resources. Ordinary user accounts, with a username and password, are created for everyone who wants to use the computer. In a multi-user operating system, like GNU/Linux, every user will have one directory inside which he can create sub-directories and files. This is called the 'home directory' of that user. Home directory of one user cannot be modified by another user. \end_layout \begin_layout Standard The operating system files are owned by \shape italic root \shape default . The /home directory contains subdirectories named after every ordinary user, for example, the user \shape italic fred \shape default owns the directory \shape italic /home/fred \shape default (fig \begin_inset LatexCommand ref reference "fig:The-GNU/Linux-file" \end_inset ) and its contents. That is also called the user's home directory. Every file and directory has three types of permissions : read, write and execute. To view them use the 'ls -l ' command. The first character of output line tells the type of the file. The next three characters show the \shape italic rwx \shape default (read, write, execute) permissions for the owner of that file. Next three for the users belonging to the same group and the next three for other users. A hyphen character (-) means the permission corresponding to that field is not granted. For example, the figure \begin_inset LatexCommand ref reference "fig:The-command-terminal" \end_inset shows a listing of five files: \end_layout \begin_layout Enumerate asecret.dat : read & write for the owner. No one else can even see it. \end_layout \begin_layout Enumerate foo.png : rw for owner, but others can view the file. \end_layout \begin_layout Enumerate hello.py : rwx for owner, others can view and execute. \end_layout \begin_layout Enumerate share.tex : rw for owner and other members of the same group. \end_layout \begin_layout Enumerate xdata is a directory. Note that execute permission is required to view contents of a directory. \end_layout \begin_layout Standard The system of ownerships and permissions also protects the system from virus attacks \begin_inset Foot status collapsed \begin_layout Standard Do not expect this from the MS Windows system. Even though it allows to create users, any user ( by running programs or viruses) is allowed to modify the system files. This may be because it grew from a single process system like MSDOS and still keeps that legacy. \end_layout \end_inset . The virus programs damage the system by modifying some application program. On a true multi-user system, for example GNU/Linux, the application program and other system files are owned by the root user and ordinary users have no permission to modify them. When a virus attempts to modify an application, it fails due to this permission and ownership scheme. \end_layout \begin_layout Subsubsection Current Directory \end_layout \begin_layout Standard There is a working directory for every user. You can create subdirectories inside that and change your current working directory to any of them. While using the command-line interface, you can use the 'cd' command to change the current working directory. Figure \begin_inset LatexCommand ref reference "fig:The-command-terminal" \end_inset shows how to change the directory and come back to the parent directory by using double dots. We also used the command 'pwd' to print the name of the current working directory. \end_layout \begin_layout Section Text Editors \end_layout \begin_layout Standard To create and modify files, we use different application programs depending on the type of document contained in that file. Text editors are used for creating and modifying plain text matter, without any formatting information embedded inside. Computer programs are plain text files and to write computer programs, we need a text editor. \shape italic 'gedit' \shape default is a simple, easy to use text editor available on GNU/Linux, which provides syntax high-lighting for several programming languages. \end_layout \begin_layout Section High Level Languages \end_layout \begin_layout Standard In order to solve a problem using a computer, it is necessary to evolve a detailed and precise step by step method of solution. A set of these precise and unambiguous steps is called an Algorithm. It should begin with steps accepting input data and should have steps which gives output data. For implementing any algorithm on a computer, each of it's steps must be converted into proper machine language instructions. Doing this process manually is called Machine Language Programming. Writing machine language programs need great care and a deep understanding about the internal structure of the computer hardware. High level languages are designed to overcome these difficulties. Using them one can create a program without knowing much about the computer hardware. \end_layout \begin_layout Standard We already learned that to solve a problem we require an algorithm and it has to be executed step by step. It is possible to express the algorithm using a set of precise and unambiguous notations. The notations selected must be suitable for the problems to be solved. \shape italic A high level programming language is a set of well defined notations which is capable of expressing algorithms. \end_layout \begin_layout Standard In general a high level language should have the following features. \end_layout \begin_layout Enumerate Ability to represent different data types like characters, integers and real numbers. In addition to this it should also support a collection of similar objects like character strings, arrays etc. \end_layout \begin_layout Enumerate Arithmetic and Logical operators that acts on the supported data types. \end_layout \begin_layout Enumerate Control flow structures for decision making, branching, looping etc. \end_layout \begin_layout Enumerate A set of syntax rules that precisely specify the combination of words and symbols permissible in the language. \end_layout \begin_layout Enumerate A set of semantic rules that assigns a single, precise and unambiguous meaning to each syntactically correct statement. \end_layout \begin_layout Standard Program text written in a high level language is often called the Source Code. It is then translated into the machine language by using translator programs. There are two types of translator programs, the Interpreter and the Compiler. \shape italic \size small Interpreter reads the high level language program line by line, translates and executes it. Compilers convert the entire program in to machine language and stores it to a file which can be executed. \end_layout \begin_layout Standard High level languages make the programming job easier. We can write programs that are machine independent. For the same program different compilers can produce machine language code to run on different types of computers and operating systems. BASIC, COBOL, FORTRAN, C, C++, Python etc. are some of the popular high level languages, each of them having advantages in different fields. \end_layout \begin_layout Standard To write any useful program for solving a problem, one has to develop an algorithm. The algorithm can be expressed in any suitable high level language. \shape italic Learning how to develop an algorithm is different from learning a programming language. \color black Learning a programming language means learning the notations, syntax and semantic rules of that language \color inherit . \shape default Best way to do this is by writing small programs with very simple algorithms. After becoming familiar with the notations and rules of the language one can start writing programs to implement more complicated algorithms. \end_layout \begin_layout Section On Free Software \end_layout \begin_layout Standard Software that can be used, studied, modified and redistributed in modified or unmodified form without restriction is called Free Software. In practice, for software to be distributed as free software, the human-readabl e form of the program (the source code) must be made available to the recipient along with a notice granting the above permissions. \end_layout \begin_layout Standard The free software movement was conceived in 1983 by Richard Stallman to give the benefit of "software freedom" to computer users. Stallman founded the Free Software Foundation in 1985 to provide the organizati onal structure to advance his Free Software ideas. Later on, alternative movements like Open Source Software came. \end_layout \begin_layout Standard Software for almost all applications is currently available under the pool of Free Software. GNU/Linux operating system, OpenOffice.org office suite, LaTeX typesetting system, Apache web server, GIMP image editor, GNU compiler collection, Python interpreter etc. are some of the popular examples. For more information refer to www.gnu.org website. \end_layout \begin_layout Section Exercises \end_layout \begin_layout Enumerate What are the basic hardware components of a computer. \end_layout \begin_layout Enumerate Name the working directory of a user named 'ramu' under GNU/Linux. \end_layout \begin_layout Enumerate What is the command to list the file names inside a directory (folder). \end_layout \begin_layout Enumerate What is the command under GNU/Linux to create a new folder. \end_layout \begin_layout Enumerate What is the command to change the working directory. \end_layout \begin_layout Enumerate Can we install more than one operating systems on a single hard disk. \end_layout \begin_layout Enumerate Name two most popular Desktop Environments for GNU/Linux. \end_layout \begin_layout Enumerate How to open a command window from the main menu of Ubuntu GNU/Linux. \end_layout \begin_layout Enumerate Explain the file ownership and permission scheme of GNU/Linux. \end_layout \begin_layout Chapter Programming in Python \end_layout \begin_layout Standard Python is a simple, high level language with a clean syntax. It offers strong support for integration with other languages and tools, comes with extensive standard libraries, and can be learned in a few days. Many Python programmers report substantial productivity gains and feel the language encourages the development of higher quality, more maintainable code. To know more visit the Python website. \begin_inset Foot status collapsed \begin_layout Standard http://www.python.org/ \end_layout \begin_layout Standard http://docs.python.org/tutorial/ \end_layout \begin_layout Standard This document, example programs and a GUI program to browse through them are at \end_layout \begin_layout Standard http://www.iuac.res.in/phoenix \end_layout \end_inset \end_layout \begin_layout Section Getting started with Python \end_layout \begin_layout Standard To start programming in Python, we have to learn how to type the source code and save it to a file, using a text editor program. We also need to know how to open a Command Terminal and start the Python Interpreter. The details of this process may vary from one system to another. On an Ubuntu GNU/Linux system, you can open the \shape italic Text Editor \shape default and the \shape italic Terminal \shape default from the Applications->Accessories menu. \end_layout \begin_layout Subsection Two modes of using Python Interpreter \end_layout \begin_layout Standard If you issue the command 'python', without any argument, from the command terminal, the Python interpreter will start and display a \shape italic '>>>' \shape default prompt where you can type Python statements. Use this method only for viewing the results of single Python statements, for example to use Python as a calculator. It could be confusing when you start writing larger programs, having looping and conditional statements. The preferred way is to enter your source code in a text editor, save it to a file (with .py extension) and execute it from the command terminal using Python. A screen-shot of the Desktop with Text Editor and Terminal is shown in figure \begin_inset LatexCommand ref reference "fig:Text-Editor-and" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/python_edit.png lyxscale 50 width 12cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Text Editor and Terminal Windows. \begin_inset LatexCommand label name "fig:Text-Editor-and" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard In this document, we will start writing small programs showing the essential elements of the language without going into the details. The reader is expected to run these example programs and also to modify them in different ways.It is like learning to drive a car, you master it by practicing. \end_layout \begin_layout Standard Let us start with a program to display the words \emph on \color black Hello World \emph default \color inherit on the computer screen. This is the customary 'hello world' program. There is another version that prints 'Goodbye cruel world', probably invented by those who give up at this point. The Python 'hello world' program is shown below. \end_layout \begin_layout Standard \align left \emph on \color black Example. hello.py \end_layout \begin_layout LyX-Code \shape italic \emph on \color black print 'Hello World' \end_layout \begin_layout Standard This should be entered into a text file using any text editor. On a GNU/Linux system you may use the text editor like 'gedit' to create the source file, save it as \shape italic \color black hello.py \shape default \color inherit . The next step is to call the Python Interpreter to execute the new program. For that, open a command terminal and (at the $ prompt) type: \begin_inset Foot status collapsed \begin_layout Standard For quick practicing, boot from the CD provided with this book and click on the learn-by-coding icon to browse through the example programs given in this book. The browser allows you to run any of them with a single click, modify and save the modified versions. \end_layout \end_inset \end_layout \begin_layout Standard $ python hello.py \end_layout \begin_layout Section Variables and Data Types \end_layout \begin_layout Standard As mentioned earlier, any high level programming language should support several data types. The problem to be solved is represented using variables belonging to the supported data types. Python supports numeric data types like integers, floating point numbers and complex numbers. To handle character strings, it uses the String data type. Python also supports other compound data types like lists, tuples, dictionaries etc. \end_layout \begin_layout Standard In languages like C, C++ and Java, we need to explicitly declare the type of a variable. This is not required in Python. The data type of a variable is decided by the value assigned to it. This is called dynamic data typing. The type of a particular variable can change during the execution of the program. If required, one type of variable can be converted in to another type by explicit type casting, like \begin_inset Formula $y=float(3)$ \end_inset . Strings are enclosed within single quotes or double quotes. \end_layout \begin_layout Standard The program \shape italic first.py \shape default shows how to define variables of different data types. It also shows how to embed comments inside a program. \end_layout \begin_layout Standard \align left \emph on \color black Example: first.py \end_layout \begin_layout LyX-Code ''' \end_layout \begin_layout LyX-Code A multi-line comment, within a pair of three single quotes. \end_layout \begin_layout LyX-Code In a line, anything after a # sign is also a comment \end_layout \begin_layout LyX-Code ''' \end_layout \begin_layout LyX-Code x = 10 \end_layout \begin_layout LyX-Code print x, type(x) # print x and its type \end_layout \begin_layout LyX-Code x = 10.4 \end_layout \begin_layout LyX-Code print x, type(x) \end_layout \begin_layout LyX-Code x = 3 + 4j \end_layout \begin_layout LyX-Code print x, type(x) \end_layout \begin_layout LyX-Code x = 'I am a String ' \end_layout \begin_layout LyX-Code print x, type(x) \end_layout \begin_layout Standard The output of the program is shown below. Note that the type of the variable \shape italic x \shape default changes during the execution of the program, depending on the value assigned to it. \end_layout \begin_layout LyX-Code 10 \end_layout \begin_layout LyX-Code 10.4 \end_layout \begin_layout LyX-Code (3+4j) \end_layout \begin_layout LyX-Code I am a String \end_layout \begin_layout Standard \align block The program treats the variables like humans treat labelled envelopes. We can pick an envelope, write some name on it and keep something inside it for future use. In a similar manner the program creates a variable, gives it a name and keeps some value inside it, to be used in subsequent steps. So far we have used four data types of Python: int, float, complex and str. To become familiar with them, you may write simple programs performing arithmetic and logical operations using them. \end_layout \begin_layout Standard \align left \emph on \color black Example: oper.py \end_layout \begin_layout LyX-Code x = 2 \end_layout \begin_layout LyX-Code y = 4 \end_layout \begin_layout LyX-Code print x + y * 2 \end_layout \begin_layout LyX-Code s = 'Hello ' \end_layout \begin_layout LyX-Code print s + s \end_layout \begin_layout LyX-Code print 3 * s \end_layout \begin_layout LyX-Code print x == y \end_layout \begin_layout LyX-Code print y == 2 * x \end_layout \begin_layout Standard Running the program \shape italic oper.py \shape default will generate the following output. \end_layout \begin_layout LyX-Code 10 \end_layout \begin_layout LyX-Code Hello Hello \end_layout \begin_layout LyX-Code Hello Hello Hello \end_layout \begin_layout LyX-Code False \end_layout \begin_layout LyX-Code True \end_layout \begin_layout Standard Note that a String can be added to another string and it can be multiplied by an integer. Try to understand the logic behind that and also try adding a String to an Integer to see what is the error message you will get. We have used the logical operator \begin_inset Formula $==$ \end_inset for comparing two variables. \end_layout \begin_layout Section Operators and their Precedence \end_layout \begin_layout Standard Python supports a large number of arithmetic and logical operators. They are summarized in the table \begin_inset LatexCommand ref reference "tab:Operators-in-Python" \end_inset . An important thing to remember is their precedence. In the expression \shape italic 2+3*4 \shape default , is the addition done first or the multiplication? According to elementary arithmetics, the multiplication should be done first. It means that the multiplication operator has higher precedence than the addition operator. If you want the addition to be done first, enforce it by using parenthesis like \begin_inset Formula $(2+3)*4$ \end_inset . Whenever there is ambiguity in evaluation, use parenthesis to clarify the order of evaluation. \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Tabular \begin_inset Text \begin_layout Standard \shape italic \size small Operator \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Description \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Expression \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Result \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small or \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Boolean OR \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 0 or 4 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 4 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small and \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Boolean AND \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 3 and 0 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 0 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small not x \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Boolean NOT \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small not 0 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small True \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small in, not in \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Membership tests \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 3 in [2.2,3,12] \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small True \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small <, <=, >, >=, !=, == \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Comparisons \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 2 > 3 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small False \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small | \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Bitwise OR \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 1 | 2 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 3 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small ^ \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Bitwise XOR \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 1 ^ 5 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 4 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small & \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Bitwise AND \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 1 & 3 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 1 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small <<, >> \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Bitwise Shifting \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 1 << 3 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 8 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small + , - \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Add, Subtract \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 6 - 4 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 2 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small *, /, % \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Multiply, divide, reminder \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 5 % 2 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 1 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small +x , -x \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Positive, Negative \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small -5*2 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small -10 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small ~ \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Bitwise NOT \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small ~1 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small -2 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small ** \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Exponentiation \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 2 ** 3 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 8 \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small x[index] \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small Subscription \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small a='abcd' ; a[1] \end_layout \end_inset \begin_inset Text \begin_layout Standard \shape italic \size small 'b' \end_layout \end_inset \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Operators in Python listed according to their precedence. \begin_inset LatexCommand label name "tab:Operators-in-Python" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Section Python Strings \end_layout \begin_layout Standard So far we have come across four data types: Integer, Float, Complex and String. Out of which, String is somewhat different from the other three. It is a collection of same kind of elements, characters. The individual elements of a String can be accessed by indexing as shown in \shape italic string.py \shape default . String is a compound, or collection, data type. \end_layout \begin_layout Standard \align left \emph on \color black Example: string.py \end_layout \begin_layout LyX-Code s = 'hello world' \end_layout \begin_layout LyX-Code print s[0] # print first element, h \end_layout \begin_layout LyX-Code print s[1] # print e \end_layout \begin_layout LyX-Code print s[-1] # will print the last character \end_layout \begin_layout Standard Addition and multiplication is defined for Strings, as demonstrated by string2.py. \end_layout \begin_layout Standard \align left \emph on \color black Example: string2.py \end_layout \begin_layout LyX-Code a = 'hello'+'world' \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code b = 'ha' * 3 \end_layout \begin_layout LyX-Code print b \end_layout \begin_layout LyX-Code print a[-1] + b[0] \end_layout \begin_layout Standard \align left will give the output \end_layout \begin_layout Standard helloworld \end_layout \begin_layout Standard hahaha \end_layout \begin_layout Standard dh \end_layout \begin_layout Standard \align left The last element of \shape italic a \shape default and first element of \shape italic b \shape default are added, resulting in the string 'dh' \end_layout \begin_layout Subsection Slicing \end_layout \begin_layout Standard Part of a String can be extracted using the slicing operation. It can be considered as a modified form of indexing a single character. Indexing using \begin_inset Formula $s[a:b]$ \end_inset extracts elements \begin_inset Formula $s[a]$ \end_inset to \begin_inset Formula $s[b-1]$ \end_inset . We can skip one of the indices. If the index on the left side of the colon is skipped, slicing starts from the first element and if the index on right side is skipped, slicing ends with the last element. \end_layout \begin_layout Standard \align left \emph on \color black Example: slice.py \end_layout \begin_layout Standard a = 'hello world' \end_layout \begin_layout Standard print a[3:5] \end_layout \begin_layout Standard print a[6:] \end_layout \begin_layout Standard print a[:5] \end_layout \begin_layout Standard \align left The reader can guess the nature of slicing operation from the output of this code, shown below. \end_layout \begin_layout LyX-Code 'lo' \end_layout \begin_layout LyX-Code 'world' \end_layout \begin_layout LyX-Code 'hello' \end_layout \begin_layout Standard Please note that specifying a right side index more than the length of the string is equivalent to skipping it. Modify \shape italic slice.py \shape default to print the result of \begin_inset Formula $a[6:20]$ \end_inset to demonstrate it. \end_layout \begin_layout Section Python Lists \end_layout \begin_layout Standard List is an important data type of Python. It is much more flexible than String. The individual elements can be of any type, even another list. Lists are defined by enclosing the elements inside a pair of square brackets, separated by commas. The program \shape italic list1.py \shape default defines a list and print its elements. \end_layout \begin_layout Standard \align left \emph on \color black Example: list1.py \end_layout \begin_layout LyX-Code a = [2.3, 3.5, 234] # make a list \end_layout \begin_layout LyX-Code print a[0] \end_layout \begin_layout LyX-Code a[1] = 'haha' # Change an element \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout Standard The output is shown below \begin_inset Foot status collapsed \begin_layout Standard The floating point number 2.3 showing as 2.2999999999999998 is interesting. This is the very nature of floting point representation of numbers, nothing to do with Python. With the precision we are using, the error in representing 2.3 is around 2.0e-16. This becomes a concern in operations like inversion of big matrices. \end_layout \end_inset . \end_layout \begin_layout Standard 2.3 \end_layout \begin_layout Standard [2.2999999999999998, 'haha', 234] \end_layout \begin_layout Standard Lists can be sliced in a manner similar to that if Strings. List addition and multiplication are demonstrated by the following example. We can also have another list as an element of a list. \end_layout \begin_layout Standard \align left \emph on \color black Example: list2.py \end_layout \begin_layout LyX-Code a = [1,2] \end_layout \begin_layout LyX-Code print a * 2 \end_layout \begin_layout LyX-Code print a + [3,4] \end_layout \begin_layout LyX-Code b = [10, 20, a] \end_layout \begin_layout LyX-Code print b \end_layout \begin_layout Standard The output of this program is shown below. \end_layout \begin_layout Standard [1, 2, 1, 2] \end_layout \begin_layout Standard [1, 2, 3, 4] \end_layout \begin_layout Standard [10, 20, [1, 2] ] \end_layout \begin_layout Section Mutable and Immutable Types \end_layout \begin_layout Standard There is one major difference between String and List types, List is mutable but String is not. \shape italic We can change the value of an element in a list, add new elements to it and remove any existing element. This is not possible with String type \shape default . Uncomment the last line of \shape italic third.py \shape default and run it to clarify this point. \end_layout \begin_layout Standard \align left \emph on \color black Example: third.py \end_layout \begin_layout LyX-Code s = [3, 3.5, 234] # make a list \end_layout \begin_layout LyX-Code s[2] = 'haha' # Change an element \end_layout \begin_layout LyX-Code print s \end_layout \begin_layout LyX-Code x = 'myname' # String type \end_layout \begin_layout LyX-Code #x[1] = 2 # uncomment to get ERROR \end_layout \begin_layout Standard The List data type is very flexible, an element of a list can be another list. We will be using lists extensively in the coming chapters. Tuple is another data type similar to List, except that it is immutable. List is defined inside square brackets, tuple is defined in a similar manner but inside parenthesis, like \begin_inset Formula $(3,3.5,234)$ \end_inset . \end_layout \begin_layout Section Input from the Keyboard \end_layout \begin_layout Standard Since most of the programs require some input from the user, let us introduce this feature before proceeding further. There are mainly two functions used for this purpose, \emph on \color black input() \emph default \color inherit for numeric type data and \emph on \color black raw_input() \emph default \color inherit for String type data. A message to be displayed can be given as an argument while calling these functions. \begin_inset Foot status collapsed \begin_layout Standard Functions will be introduced later. For the time being, understand that it is an isolated piece of code, called from the main program with some input arguments and returns some output. \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on \color black Example: kin1.py \end_layout \begin_layout LyX-Code x = input('Enter an integer ') \end_layout \begin_layout LyX-Code y = input('Enter one more ') \end_layout \begin_layout LyX-Code print 'The sum is ', x + y \end_layout \begin_layout LyX-Code s = raw_input('Enter a String ') \end_layout \begin_layout LyX-Code print 'You entered ', s \end_layout \begin_layout Standard It is also possible to read more than one variable using a single input() statement. \emph on \color black String \emph default \color inherit type data read using raw_input() may be converted into \emph on \color black integer \emph default \color inherit or \emph on \color black float \emph default \color inherit type if they contain only the valid characters. In order to show the effect of conversion explicitly, we multiply the variables by 2 before printing. Multiplying a String by 2 prints it twice. If the String contains any other characters than \shape italic 0..9, . and e \shape default , the conversion to float will give an error. \end_layout \begin_layout Standard \align left \emph on \color black Example: kin2.py \end_layout \begin_layout LyX-Code x,y = input('Enter x and y separated by comma ') \end_layout \begin_layout LyX-Code print 'The sum is ', x + y \end_layout \begin_layout LyX-Code s = raw_input('Enter a decimal number ') \end_layout \begin_layout LyX-Code a = float(s) \end_layout \begin_layout LyX-Code print s * 2 # prints string twice \end_layout \begin_layout LyX-Code print a * 2 # converted value times 2 \end_layout \begin_layout Standard We have learned about the basic data types of Python and how to get input data from the keyboard. This is enough to try some simple problems and algorithms to solve them. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit area.py \end_layout \begin_layout LyX-Code pi = 3.1416 \end_layout \begin_layout LyX-Code r = input('Enter Radius ') \end_layout \begin_layout LyX-Code a = pi * r ** 2 # \begin_inset Formula $A=\pi r^{2}$ \end_inset \end_layout \begin_layout LyX-Code print 'Area = ', a \end_layout \begin_layout Standard \align block The above example calculates the area of a circle. Line three calculates \begin_inset Formula $r^{2}$ \end_inset \InsetSpace ~ using the exponentiation operator \begin_inset Formula $**$ \end_inset , and multiply it with \begin_inset Formula $\pi$ \end_inset using the multiplication operator \begin_inset Formula $*$ \end_inset . \begin_inset Formula $r^{2}$ \end_inset is evaluated first because ** has higher precedence than *, otherwise the result would be \begin_inset Formula $(\pi r)^{2}$ \end_inset . \end_layout \begin_layout Section Iteration: while and for loops \end_layout \begin_layout Standard If programs can only execute from the first line to the last in that order, as shown in the previous examples, it would be impossible to write any useful program. For example, we need to print the multiplication table of eight. Using our present knowledge, it would look like the following \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit badtable.py \end_layout \begin_layout LyX-Code print 1 * 8 \end_layout \begin_layout LyX-Code print 2 * 8 \end_layout \begin_layout LyX-Code print 3 * 8 \end_layout \begin_layout LyX-Code print 4 * 8 \end_layout \begin_layout LyX-Code print 5 * 8 \end_layout \begin_layout Standard Well, we are stopping here and looking for a better way to do this job. \end_layout \begin_layout Standard The solution is to use the \emph on \color black while \emph default \color inherit loop of Python. The logical expression in front of \shape italic while \shape default is evaluated, and if it is True, the body of the while loop (the indented lines below the while statement) is executed. The process is repeated until the condition becomes false. We should have some statement inside the body of the loop that will make this condition false after few iterations. Otherwise the program will run in an infinite loop and you will have to press Control-C to terminate it. \end_layout \begin_layout Standard The program \shape italic table.py \shape default , defines a variable \begin_inset Formula $x$ \end_inset and assigns it an initial value of 1. Inside the while loop \begin_inset Formula $x*8$ \end_inset is printed and the value of \begin_inset Formula $x$ \end_inset is incremented. This process will be repeated until the value of \begin_inset Formula $x$ \end_inset becomes greater than 10. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit table.py \end_layout \begin_layout LyX-Code x = 1 \end_layout \begin_layout LyX-Code while x <= 10: \end_layout \begin_layout LyX-Code print x * 8 \end_layout \begin_layout LyX-Code x = x + 1 \end_layout \begin_layout Standard As per the Python syntax, the while statement ends with a colon and the code inside the \emph on \color black while \emph default \color inherit loop is indented. Indentation can be done using tab or few spaces. In this example, we have demonstrated a simple algorithm. \end_layout \begin_layout Subsection Python Syntax, Colon & Indentation \end_layout \begin_layout Standard Python was designed to be a highly readable language. It has a relatively uncluttered visual layout, uses English keywords frequently where other languages use punctuation, and has notably fewer syntactic constructions than other popular structured languages. \end_layout \begin_layout Standard There are mainly two things to remember about Python syntax: \shape italic indentation and colon \shape default . \shape italic Python uses indentation to delimit blocks of code \shape default . Both space characters and tab characters are currently accepted as forms of indentation in Python. Mixing spaces and tabs can create bugs that are hard to find, since the text editor does not show the difference. There should not be any extra white spaces in the beginning of any line. \end_layout \begin_layout Standard \shape italic The line before any indented block must end with a colon character \shape default . \end_layout \begin_layout Subsection Syntax of 'for loops' \end_layout \begin_layout Standard Python \shape italic for \shape default loops are slightly different from the for loops of other languages. Python \shape italic for \shape default loop iterates over a compound data type like a String, List or Tuple. During each iteration, one member of the compound data is assigned to the loop variable. The flexibility of this can be seen from the examples below. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit forloop.py \end_layout \begin_layout LyX-Code a = 'Hello' \end_layout \begin_layout LyX-Code for ch in a: # ch is the loop variable \end_layout \begin_layout LyX-Code print ch \end_layout \begin_layout LyX-Code b = ['haha', 3.4, 2345, 3+5j] \end_layout \begin_layout LyX-Code for item in b: \end_layout \begin_layout LyX-Code print item \end_layout \begin_layout Standard which gives the output : \end_layout \begin_layout Standard H \end_layout \begin_layout Standard e \end_layout \begin_layout Standard l \end_layout \begin_layout Standard l \end_layout \begin_layout Standard o \end_layout \begin_layout Standard haha \end_layout \begin_layout Standard 3.4 \end_layout \begin_layout Standard 2345 \end_layout \begin_layout Standard (3+5j) \end_layout \begin_layout Standard For constructing for loops that executes a fixed number of times, we can create a list using the range() function and run the for loop over that. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit forloop2.py \end_layout \begin_layout LyX-Code mylist = range(5) \end_layout \begin_layout LyX-Code print mylist \end_layout \begin_layout LyX-Code for item in mylist: \end_layout \begin_layout LyX-Code print item \end_layout \begin_layout Standard The output will look like : \end_layout \begin_layout Standard [0, 1, 2, 3, 4] \end_layout \begin_layout Standard 0 \end_layout \begin_layout Standard 1 \end_layout \begin_layout Standard 2 \end_layout \begin_layout Standard 3 \end_layout \begin_layout Standard 4 \end_layout \begin_layout Standard The range function in the above example generates the list \begin_inset Formula $[0,1,2,3,4]$ \end_inset and the for loop walks thorugh it printing each member. It is possible to specify the starting point and increment as arguments in the form range(start, end+1, step). The following example prints the table of 5 using this feature. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit forloop3.py \end_layout \begin_layout LyX-Code mylist = range(5,51,5) \end_layout \begin_layout LyX-Code for item in mylist: \end_layout \begin_layout LyX-Code print item , \end_layout \begin_layout Standard The output is shown below. \end_layout \begin_layout Standard 5 10 15 20 25 30 35 40 45 50 \end_layout \begin_layout Standard The print statement inserts a newline at the end by default. We can suppress this behaviour by adding a comma character at the end as done in the previous example. \end_layout \begin_layout Standard In some cases, we may need to traverse the list to modify some or all of the elements. This can be done by looping over a list of indices generated by the range() function.For example, the program \emph on \emph default forloop4.py \size large \emph on \size default \emph default zeros all the elements of the list. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit forloop4.py \end_layout \begin_layout LyX-Code a = [2, 5, 3, 4, 12] \end_layout \begin_layout LyX-Code size = len(a) \end_layout \begin_layout LyX-Code for k in range(size): \end_layout \begin_layout LyX-Code a[k] = 0 \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout Section Conditional Execution: if, elif and else \end_layout \begin_layout Standard In some cases, we may need to execute some section of the code only if certain conditions are true. Python implements this feature using the \emph on \color black if, elif \emph default \color inherit and \emph on \color black else \emph default \color inherit keywords, as shown in the next example. The indentation levels of \shape italic if \shape default and the corresponding \shape italic elif \shape default and \shape italic else \shape default must be kept the same. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit compare.py \end_layout \begin_layout LyX-Code x = raw_input('Enter a string ') \end_layout \begin_layout LyX-Code if x == 'hello': \end_layout \begin_layout LyX-Code print 'You typed ', x \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit big.py \end_layout \begin_layout LyX-Code x = input('Enter a number ') \end_layout \begin_layout LyX-Code if x > 10: \end_layout \begin_layout LyX-Code print 'Bigger Number' \end_layout \begin_layout LyX-Code elif x < 10: \end_layout \begin_layout LyX-Code print 'Smaller Number' \end_layout \begin_layout LyX-Code else: \end_layout \begin_layout LyX-Code print 'Same Number' \end_layout \begin_layout Standard The statement \shape italic x > 10 and x < 15 \shape default can be expressed in a short form, like \shape italic 10 < x < 15 \shape default . \end_layout \begin_layout Standard The next example uses \emph on \color black while \emph default \color inherit and \emph on \color black if \emph default \color inherit keywords in the same program. Note the level of indentation when the if statement comes inside the while loop. Remember that, the \shape italic if \shape default statement must be aligned with the corresponding \shape italic elif and else. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit big2.py \end_layout \begin_layout LyX-Code x = 1 \end_layout \begin_layout LyX-Code while x < 11: \end_layout \begin_layout LyX-Code if x < 5: \end_layout \begin_layout LyX-Code print 'Small ', x \end_layout \begin_layout LyX-Code else: \end_layout \begin_layout LyX-Code print 'Big ', x \end_layout \begin_layout LyX-Code x = x + 1 \end_layout \begin_layout LyX-Code print 'Done' \end_layout \begin_layout Section Modify loops : break and continue \end_layout \begin_layout Standard We can use the \emph on \color black break \emph default \color inherit statement to terminate a loop, if some condition is met. The \shape italic continue \shape default statement is used to skip the rest of the block and go to the beginning again. Both are demonstrated in the program \shape italic big3.py \shape default shown below. \end_layout \begin_layout Standard \align left \emph on \color black Example: \emph default \color inherit big3.py \end_layout \begin_layout LyX-Code x = 1 \end_layout \begin_layout LyX-Code while x < 10: \end_layout \begin_layout LyX-Code if x < 3: \end_layout \begin_layout LyX-Code print 'skipping work', x \end_layout \begin_layout LyX-Code x = x + 1 \end_layout \begin_layout LyX-Code continue \end_layout \begin_layout LyX-Code print x \end_layout \begin_layout LyX-Code if x == 4: \end_layout \begin_layout LyX-Code print 'Enough of work' \end_layout \begin_layout LyX-Code break \end_layout \begin_layout LyX-Code x = x + 1 \end_layout \begin_layout LyX-Code print 'Done' \end_layout \begin_layout Standard The output of big3.py is listed below. \end_layout \begin_layout Standard skipping work 1 \end_layout \begin_layout Standard skipping work 2 \end_layout \begin_layout Standard 3 \end_layout \begin_layout Standard 4 \end_layout \begin_layout Standard Enough of work \end_layout \begin_layout Standard Done \end_layout \begin_layout Standard \align block Now let us write a program to find out the largest positive number entered by the user. The algorithm works in the following manner. To start with, we assume that the largest number is zero. After reading a number, the program checks whether it is bigger than the current value of the largest number. If so the value of the largest number is replaced with the current number. The program terminates when the user enters zero. Modify max.py to work with negative numbers also. \end_layout \begin_layout Standard \align left Example: max.py \end_layout \begin_layout LyX-Code max = 0 \end_layout \begin_layout LyX-Code while True: # Infinite loop \end_layout \begin_layout LyX-Code x = input('Enter a number ') \end_layout \begin_layout LyX-Code if x > max: \end_layout \begin_layout LyX-Code max = x \end_layout \begin_layout LyX-Code if x == 0: \end_layout \begin_layout LyX-Code print max \end_layout \begin_layout LyX-Code break \end_layout \begin_layout Section Line joining \end_layout \begin_layout Standard Python interpreter processes the code line by line. A program may have a long line of code that may not physically fit in the width of the text editor. In such cases, we can split a logical line of code into more than one physical lines, using backslash characters ( \backslash ), in other words multiple physical lines are joined to form a logical line before interpreting it. \end_layout \begin_layout LyX-Code if 1900 < year < 2100 and 1 <= month <= 12 : \end_layout \begin_layout Standard can be split like \end_layout \begin_layout LyX-Code if 1900 < year < 2100 \backslash \end_layout \begin_layout LyX-Code and 1 <= month <= 12 : \end_layout \begin_layout Standard Do not split in the middle of words except for Strings. A long String can be split as shown below. \end_layout \begin_layout LyX-Code longname = 'I am so long and will \backslash \end_layout \begin_layout LyX-Code not fit in a single line' \end_layout \begin_layout LyX-Code print longname \end_layout \begin_layout Section Exercises \end_layout \begin_layout Standard We have now covered the minimum essentials of Python; defining variables, performing arithmetic and logical operations on them and the control flow statements. These are sufficient for handling most of the programming tasks. It would be better to get a grip of it before proceeding further, by writing some code. \end_layout \begin_layout Enumerate Modify the expression \shape italic print 5+3*2 \shape default to get a result of 16 \end_layout \begin_layout Enumerate What will be the output of \shape italic print type(4.5) \end_layout \begin_layout Enumerate Print all even numbers upto 30, suffixed by a * if the number is a multiple of 6. (hint: use % operator) \end_layout \begin_layout Enumerate Write Python code to remove the last two characters of 'I am a long string' by slicing, without counting the characters. (hint: use negative indexing) \end_layout \begin_layout Enumerate s = '012345' . (a) Slice it to remove last two elements (b) remove first two element. \end_layout \begin_layout Enumerate a = [1,2,3,4,5]. Use Slicing and multiplication to generate [2,3,4,2,3,4] from it. \end_layout \begin_layout Enumerate Compare the results of 5/2, 5.0/2 and 2.0/3. \end_layout \begin_layout Enumerate Print the following pattern using a while loop \newline + \newline ++ \newline +++ \newline ++++ \end_layout \begin_layout Enumerate Write a program to read inputs like 8A, 10C etc. and print the integer and alphabet parts separately. \end_layout \begin_layout Enumerate Write code to print a number in the binary format (for example 5 will be printed as 101) \end_layout \begin_layout Enumerate Write code to print all perfect cubes upto 2000. \end_layout \begin_layout Enumerate Write a Python program to print the multiplication table of 5. \end_layout \begin_layout Enumerate Write a program to find the volume of a box with sides 3,4 and 5 inches in \begin_inset Formula $cm^{3}$ \end_inset ( 1 inch = 2.54 cm) \end_layout \begin_layout Enumerate Write a program to find the percentage of volume occupied by a sphere of diameter \begin_inset Formula $r$ \end_inset fitted in a cube of side \begin_inset Formula $r$ \end_inset . Read \begin_inset Formula $r$ \end_inset from the keyboard. \end_layout \begin_layout Enumerate Write a Python program to calculate the area of a circle. \end_layout \begin_layout Enumerate Write a program to divide an integer by another without using the / operator. (hint: use - operator) \end_layout \begin_layout Enumerate Count the number of times the character 'a' appears in a String read from the keyboard. Keep on prompting for the string until there is no 'a' in the input. \end_layout \begin_layout Enumerate Create an integer division machine that will ask the user for two numbers then divide and give the result. The program should give the result in two parts: the whole number result and the remainder. Example: If a user enters 11 / 4, the computer should give the result 2 and remainder 3. \end_layout \begin_layout Enumerate Modify the previous program to avoid division by zero error. \end_layout \begin_layout Enumerate Create an adding machine that will keep on asking the user for numbers, add them together and show the total after each step. Terminate when user enters a zero. \end_layout \begin_layout Enumerate Modify the adding machine to use raw_input() and check for errors like user entering invalid characters. \end_layout \begin_layout Enumerate Create a script that will convert Celsius to Fahrenheit. The program should ask the users to enter the temperature in Celsius and should print out the temperature in Fahrenheit, using \begin_inset Formula $f=\frac{9}{5}c+32$ \end_inset . \end_layout \begin_layout Enumerate Write a program to convert Fahrenheit to Celsius. \end_layout \begin_layout Enumerate Create a script that uses a variable and will write 20 times "I will not talk in class." Make each sentence on a separate line. \end_layout \begin_layout Enumerate Define \begin_inset Formula $2+5j$ \end_inset and \begin_inset Formula $2-5j$ \end_inset as complex numbers , and find their product. Verify the result by defining the real and imaginary parts separately and using the multiplication formula. \end_layout \begin_layout Enumerate Write the multiplication table of 12 using while loop. \end_layout \begin_layout Enumerate Write the multiplication table of a number, from the user, using for loop. \end_layout \begin_layout Enumerate Print the powers of 2 up to 1024 using a for loop. (only two lines of code) \end_layout \begin_layout Enumerate Define the list a = [123, 12.4, 'haha', 3.4] \newline a) print all members using a for loop \newline b) print the float type members ( use type() function) \newline c) insert a member after 12.4 \newline d) append more members \end_layout \begin_layout Enumerate Make a list containing 10 members using a for loop. \end_layout \begin_layout Enumerate Generate multiplication table of 5 with two lines of Python code. (hint: range function) \end_layout \begin_layout Enumerate Write a program to find the sum of five numbers read from the keyboard. \end_layout \begin_layout Enumerate Write a program to read numbers from the keyboard until their sum exceeds 200. Modify the program to ignore numbers greater than 99. \end_layout \begin_layout Enumerate Write a Python function to calculate the GCD of two numbers \end_layout \begin_layout Enumerate Write a Python program to find annual compound interest. Get P,N and R from user \end_layout \begin_layout Section Functions \end_layout \begin_layout Standard Large programs need to be divided into small logical units. A function is generally an isolated unit of code that has a name and does a well defined job. A function groups a number of program statements into a unit and gives it a name. This unit can be invoked from other parts of a program. Python allows you to define functions using the \shape italic def \shape default keyword. A function may have one or more variables as arguments, which receive their values from the calling program. \end_layout \begin_layout Standard In the example shown below, function arguments (a and b) get the values 3 and 4 respectively from the caller. One can specify more than one variables in the return statement, separated by commas. The function will return a tuple containing those variables. Some functions may not have any arguments, but while calling them we need to use an empty parenthesis, otherwise the function will not be invoked. If there is no return statement, a None is returned to the caller. \end_layout \begin_layout Standard \align left \emph on Example func.py \end_layout \begin_layout LyX-Code def sum(a,b): # a trivial function \end_layout \begin_layout LyX-Code return a + b \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print sum(3, 4) \end_layout \begin_layout Standard The function \shape italic factorial.py \shape default calls itself recursively. The value of argument is decremented before each call. Try to understand the working of this by inserting print statements inside the function. \end_layout \begin_layout Standard \align left \emph on Example factor.py \end_layout \begin_layout LyX-Code def factorial(n): # a recursive function \end_layout \begin_layout LyX-Code if n == 0: \end_layout \begin_layout LyX-Code return 1 \end_layout \begin_layout LyX-Code else: \end_layout \begin_layout LyX-Code return n * factorial(n-1) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print factorial(10) \end_layout \begin_layout Standard \align left \emph on Example fibanocci.py \end_layout \begin_layout LyX-Code def fib(n): # print Fibonacci series up to n \end_layout \begin_layout LyX-Code a, b = 0, 1 \end_layout \begin_layout LyX-Code while b < n: \end_layout \begin_layout LyX-Code print b \end_layout \begin_layout LyX-Code a, b = b, a+b \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print fib(30) \end_layout \begin_layout Standard Runing \shape italic fibanocci.py \shape default will print \end_layout \begin_layout Quotation 1 1 2 3 5 8 13 21 \end_layout \begin_layout Standard Modify it to replace \begin_inset Formula $a,b=b,a+b$ \end_inset by two separate assignment statements, if required introduce a third variable. \end_layout \begin_layout Subsection Scope of variables \end_layout \begin_layout Standard The variables defined inside a function are not known outside the function. There could be two variables, one inside and one outside, with the same name. The program \shape italic scope.py \shape default demonstrates this feature. \end_layout \begin_layout Standard \align left \emph on Example scope.py \end_layout \begin_layout LyX-Code def change(x): \end_layout \begin_layout LyX-Code counter = x \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code counter = 10 \end_layout \begin_layout LyX-Code change(5) \end_layout \begin_layout LyX-Code print counter \end_layout \begin_layout Standard The program will print 10 and not 5. The two variables, both named counter, are not related to each other. In some cases, it may be desirable to allow the function to change some external variable. This can be achieved by using the \shape italic global \shape default keyword, as shown in \shape italic global.py \shape default . \end_layout \begin_layout Standard \align left \emph on Example global.py \end_layout \begin_layout LyX-Code def change(x): \end_layout \begin_layout LyX-Code global counter # use the global variable \end_layout \begin_layout LyX-Code counter = x \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code counter = 10 \end_layout \begin_layout LyX-Code change(5) \end_layout \begin_layout LyX-Code print counter \end_layout \begin_layout Standard The program will now print 5. Functions with global variables should be used carefully to avoid inadvertent side effects. \end_layout \begin_layout Subsection Optional and Named Arguments \end_layout \begin_layout Standard Python allows function arguments to have default values; if the function is called without a particular argument, its default value will be taken. Due to this feature, the same function can be called with different number of arguments. The arguments without default values must appear first in the argument list and they cannot be omitted while invoking the function. The following example shows a function named power() that does exponentiation, but the default value of exponent is set to 2. \end_layout \begin_layout Standard \align left \emph on Example power.py \end_layout \begin_layout LyX-Code def power(mant, exp = 2.0): \end_layout \begin_layout LyX-Code return mant ** exp \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print power(5., 3) \end_layout \begin_layout LyX-Code print power(4.) # prints 16 \end_layout \begin_layout LyX-Code print power() # Gives Error \end_layout \begin_layout Standard Arguments can be specified in any order by using named arguments. \end_layout \begin_layout Standard \align left \emph on Example named.py \end_layout \begin_layout LyX-Code def power(mant = 10.0, exp = 2.0): \end_layout \begin_layout LyX-Code return mant ** exp \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print power(5., 3) \end_layout \begin_layout LyX-Code print power(4.) # prints 16 \end_layout \begin_layout LyX-Code print power(exp=3) # mant gets 10.0, prints 1000 \end_layout \begin_layout Section More on Strings and Lists \end_layout \begin_layout Standard Before proceeding further, we will explore some of the functions provided for manipulating strings and lists. Python strings can be manipulated in many ways. The following program prints the length of a string, makes an upper case version for printing and prints a help message on the String class. \end_layout \begin_layout Standard \align left \emph on \color black Example: stringhelp.py \end_layout \begin_layout LyX-Code s = 'hello world' \end_layout \begin_layout LyX-Code print len(s) \end_layout \begin_layout LyX-Code print s.upper() \end_layout \begin_layout LyX-Code help(str) # press q to exit help \end_layout \begin_layout Standard Python is an object oriented language and all variables are objects belonging to various classes. The method upper() (a function belonging to a class is called a method) is invoked using the dot operator. All we need to know at this stage is that there are several methods that can be used for manipulating objects and they can be invoked like: \shape italic variable_name.method_name() \shape default . \end_layout \begin_layout Subsection split and join \end_layout \begin_layout Standard \align left Splitting a String will result in a list of smaller strings. If you do not specify the separator, the space character is assumed by default. To demonstrate the working of these functions, few lines of code and its output are listed below. \end_layout \begin_layout Standard \align left \emph on \color black Example: split.py \end_layout \begin_layout LyX-Code s = 'I am a long string' \end_layout \begin_layout LyX-Code print s.split() \end_layout \begin_layout LyX-Code a = 'abc.abc.abc' \end_layout \begin_layout LyX-Code aa = a.split('.') \end_layout \begin_layout LyX-Code print aa \end_layout \begin_layout LyX-Code mm = '+'.join(aa) \end_layout \begin_layout LyX-Code print mm \end_layout \begin_layout Standard \align left The result is shown below \end_layout \begin_layout Standard ['I', 'am', 'a', 'long', 'string'] \end_layout \begin_layout Standard ['abc', 'abc', 'abc'] \end_layout \begin_layout Standard 'abc+abc+abc' \end_layout \begin_layout Standard \align left The List of strings generated by split is joined using '+' character, resulting in the last line of the output. \end_layout \begin_layout Subsection Manipulating Lists \end_layout \begin_layout Standard Python lists are very flexible, we can append, insert, delete and modify elements of a list. The program \shape italic list3.py \shape default demonstrates some of them. \end_layout \begin_layout Standard \align left \emph on \color black Example: list3.py \end_layout \begin_layout LyX-Code a = [] # make an empty list \end_layout \begin_layout LyX-Code a.append(3) # Add an element \end_layout \begin_layout LyX-Code a.insert(0,2.5) # insert 2.5 as first element \end_layout \begin_layout LyX-Code print a, a[0] \end_layout \begin_layout LyX-Code print len(a) \end_layout \begin_layout Standard The output is shown below. \end_layout \begin_layout Standard [2.5, 3] 2.5 \end_layout \begin_layout Standard 2 \end_layout \begin_layout Subsection Copying Lists \end_layout \begin_layout Standard Lists cannot be copied like numeric data types. The statement \begin_inset Formula $b=a$ \end_inset will not create a new list b from list a, it just make a reference to a. The following example will clarify this point. To make a duplicate copy of a list, we need to use the \shape italic copy \shape default module. \end_layout \begin_layout Standard \align left \emph on \color black Example: list_copy.py \end_layout \begin_layout LyX-Code a = [1,2,3,4] \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code b = a # b refers to a \end_layout \begin_layout LyX-Code print a == b # True \end_layout \begin_layout LyX-Code b[0] = 5 # Modifies a[0] \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code import copy \end_layout \begin_layout LyX-Code c = copy.copy(a) \end_layout \begin_layout LyX-Code c[1] = 100 \end_layout \begin_layout LyX-Code print a is c # is False \end_layout \begin_layout LyX-Code print a, c \end_layout \begin_layout Standard The output is shown below. \end_layout \begin_layout Standard [1, 2, 3, 4] \end_layout \begin_layout Standard True \end_layout \begin_layout Standard [5, 2, 3, 4] \end_layout \begin_layout Standard False \end_layout \begin_layout Standard [5, 2, 3, 4] [5, 100, 3, 4] \end_layout \begin_layout Section Python Modules and Packages \begin_inset Foot status collapsed \begin_layout Standard While giving names to your Python programs, make sure that you are not directly or indirectly importing any Python module having same name. For example, if you create a program named \emph on math.py \emph default and keep it in your working directory, the \emph on import math \emph default statement from any other program started from that directory will try to import your file named \emph on math.py \emph default and give error. If you ever do that by mistake, delete all the files with .pyc extension from your directory. \end_layout \end_inset \end_layout \begin_layout Standard One of the major advantages of Python is the availability of libraries for various applications like graphics, networking and scientific computation. The standard library distributed with Python itself has a large number of modules: time, random, pickle, system etc. are some of them. The site http://docs.python.org/library/ has the complete reference. \end_layout \begin_layout Standard Modules are loaded by using the \shape italic import \shape default keyword. Several ways of using \shape italic import \shape default is explained below, using the math (containing mathematical functions) module as an example. \end_layout \begin_layout Subsection Different ways to import \end_layout \begin_layout Standard simplest way to use import is shown in \shape italic mathsin.py \shape default , where the function is invoked using the form \shape italic module_name.function_name() \shape default . In the next example, we use an alias for the module name. \end_layout \begin_layout Standard \align left \emph on Example mathsin.py \end_layout \begin_layout LyX-Code import math \end_layout \begin_layout LyX-Code print math.sin(0.5) # module_name.method_name \end_layout \begin_layout Standard \align left \emph on Example mathsin2.py \end_layout \begin_layout LyX-Code import math as m # Give another name for math \end_layout \begin_layout LyX-Code print m.sin(0.5) # Refer by the new name \end_layout \begin_layout Standard We can also import the functions to behave like local (like the ones within our source file) function, as shown below. The character * is a wild card for importing all the functions. \end_layout \begin_layout Standard \align left \emph on Example mathlocal.py \end_layout \begin_layout LyX-Code from math import sin # sin is imported as local \end_layout \begin_layout LyX-Code print sin(0.5) \end_layout \begin_layout Standard \align left \emph on Example mathlocal2.py \end_layout \begin_layout LyX-Code from math import * # import everything from math \end_layout \begin_layout LyX-Code print sin(0.5) \end_layout \begin_layout Standard In the third and fourth cases, we need not type the module name every time. But there could be trouble if two modules imported contains a function with same name. In the program \emph on conflict.py \emph default , the \begin_inset Formula $\sin()$ \end_inset from \emph on numpy \emph default is capable of handling a list argument. After importing \shape italic math.py \shape default , line 4, the \begin_inset Formula $\sin$ \end_inset function from \shape italic math \shape default module replaces the one from \emph on numpy \emph default . The error occurs because the \begin_inset Formula $\sin()$ \end_inset from \emph on math \emph default can accept only a numeric type argument. \end_layout \begin_layout Standard \align left \emph on Example conflict.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code x = [0.1, 0.2, 0.3] \end_layout \begin_layout LyX-Code print sin(x) # numpy's sin can handle lists \end_layout \begin_layout LyX-Code from math import * # sin of math becomes effective \end_layout \begin_layout LyX-Code print sin(x) # will give ERROR \end_layout \begin_layout Subsection Packages \end_layout \begin_layout Standard Packages are used for organizing multiple modules. The module name A.B designates a submodule named B in a package named A. The concept is demonstrated using the packages Numpy \begin_inset Foot status collapsed \begin_layout Standard NumPy will be discusssed later in chapter \begin_inset LatexCommand ref reference "sec:Arrays-and-Matrices" \end_inset . \end_layout \end_inset and Scipy. \end_layout \begin_layout Standard \align left \emph on Example submodule.py \end_layout \begin_layout LyX-Code import numpy \end_layout \begin_layout LyX-Code print numpy.random.normal() \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code import scipy.special \end_layout \begin_layout LyX-Code print scipy.special.j0(.1) \end_layout \begin_layout Standard In this example \shape italic random \shape default is a module inside the package \shape italic NumPy \shape default . Similarly \shape italic special \shape default is a module inside the package \shape italic Scipy. \shape default We use both of them in the package.module.function() format. But there is some difference. In the case of Numpy, the random module is loaded by default, importing scipy does not import the module special by default. This behavior can be defined while writing the Package and it is upto the package author. \end_layout \begin_layout Section File Input/Output \end_layout \begin_layout Standard Disk files can be opened using the function named open() that returns a File object. Files can be opened for reading or writing. There are several methods belonging to the File class that can be used for reading and writing data. \end_layout \begin_layout Standard \align left \emph on Example wfile.py \end_layout \begin_layout LyX-Code f = open('test.dat' , 'w') \end_layout \begin_layout LyX-Code f.write ('This is a test file') \end_layout \begin_layout LyX-Code f.close() \end_layout \begin_layout Standard Above program creates a new file named 'test.dat' (any existing file with the same name will be deleted) and writes a String to it. The following program opens this file for reading the data. \end_layout \begin_layout Standard \align left \emph on Example rfile.py \end_layout \begin_layout LyX-Code f = open('test.dat' , 'r') \end_layout \begin_layout LyX-Code print f.read() \end_layout \begin_layout LyX-Code f.close() \end_layout \begin_layout Standard Note that the data written/read are character strings. read() function can also be used to read a fixed number of characters, as shown below. \end_layout \begin_layout Standard \align left \emph on Example rfile2.py \end_layout \begin_layout LyX-Code f = open('test.dat' , 'r') \end_layout \begin_layout LyX-Code print f.read(7) # get first seven characters \end_layout \begin_layout LyX-Code print f.read() # get the remaining ones \end_layout \begin_layout LyX-Code f.close() \end_layout \begin_layout Standard Now we will examine how to read a text data from a file and convert it into numeric type. First we will create a file with a column of numbers. \end_layout \begin_layout Standard \align left \emph on Example wfile2.py \end_layout \begin_layout LyX-Code f = open('data.dat' , 'w') \end_layout \begin_layout LyX-Code for k in range(1,4): \end_layout \begin_layout LyX-Code s = '%3d \backslash n' %(k) \end_layout \begin_layout LyX-Code f.write(s) \end_layout \begin_layout LyX-Code f.close() \end_layout \begin_layout Standard The contents of the file created will look like this. \end_layout \begin_layout Standard 1 \end_layout \begin_layout Standard 2 \end_layout \begin_layout Standard 3 \end_layout \begin_layout Standard \align left Now we write a program to read this file, line by line, and convert the string type data to integer type, and print the numbers. \begin_inset Foot status collapsed \begin_layout Standard This will give error if there is a blank line in the data file. This can be corrected by changing the comparison statement to if \shape italic len(s) < 1: \shape default , so that the processing stops at a blank line. Modify the code to skip a blank line instead of exiting (hint: use continue ). \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example rfile3.py \end_layout \begin_layout LyX-Code f = open('data.dat' , 'r') \end_layout \begin_layout LyX-Code while 1: # infinite loop \end_layout \begin_layout LyX-Code s = f.readline() \end_layout \begin_layout LyX-Code if s == '' : # Empty string means end of file \end_layout \begin_layout LyX-Code break # terminate the loop \end_layout \begin_layout LyX-Code m = int(s) # Convert to integer \end_layout \begin_layout LyX-Code print m * 5 \end_layout \begin_layout LyX-Code f.close() \end_layout \begin_layout Subsection The pickle module \end_layout \begin_layout Standard Strings can easily be written to and read from a file. Numbers take a bit more effort, since the read() method only returns Strings, which will have to be converted in to a number explicitly. However, when you want to save and restore data types like lists, dictionaries, or class instances, things get a lot more complicated. Rather than have the users constantly writing and debugging code to save complicated data types, Python provides a standard module called pickle. \end_layout \begin_layout Standard \align left \emph on Example pickledump.py \end_layout \begin_layout LyX-Code import pickle \end_layout \begin_layout LyX-Code f = open('test.pck' , 'w') \end_layout \begin_layout LyX-Code pickle.dump(12.3, f) # write a float type \end_layout \begin_layout LyX-Code f.close() \end_layout \begin_layout Standard \align left Now write another program to read it back from the file and check the data type. \end_layout \begin_layout Standard \align left \emph on Example pickleload.py \end_layout \begin_layout LyX-Code import pickle \end_layout \begin_layout LyX-Code f = open('test.pck' , 'r') \end_layout \begin_layout LyX-Code x = pickle.load(f) \end_layout \begin_layout LyX-Code print x , type(x) # check the type of data read \end_layout \begin_layout LyX-Code f.close() \end_layout \begin_layout Section Formatted Printing \end_layout \begin_layout Standard Formatted printing is done by using a format string followed by the % operator and the values to be printed. If format requires a single argument, values may be a single variable. Otherwise, values must be a tuple (just place them inside parenthesis, separated by commas) with exactly the number of items specified by the format string. \end_layout \begin_layout Standard \align left \emph on Example: format.py \end_layout \begin_layout LyX-Code a = 2.0 /3 # 2/3 will print zero \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code print 'a = %5.3f' %(a) # upto 3 decimal places \end_layout \begin_layout Standard Data can be printed in various formats. The conversion types are summarized in the following table. There are several flags that can be used to modify the formatting, like justification, filling etc. \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Standard \align center \begin_inset Tabular \begin_inset Text \begin_layout Standard Conversion \end_layout \end_inset \begin_inset Text \begin_layout Standard Conversion \end_layout \end_inset \begin_inset Text \begin_layout Standard Example \end_layout \end_inset \begin_inset Text \begin_layout Standard Result \end_layout \end_inset \begin_inset Text \begin_layout Standard d , i \end_layout \end_inset \begin_inset Text \begin_layout Standard signed Integer \end_layout \end_inset \begin_inset Text \begin_layout Standard '%6d'%(12) \end_layout \end_inset \begin_inset Text \begin_layout Standard '\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ 12' \end_layout \end_inset \begin_inset Text \begin_layout Standard f \end_layout \end_inset \begin_inset Text \begin_layout Standard floating point decimal \end_layout \end_inset \begin_inset Text \begin_layout Standard '%6.4f'%(2.0/3) \end_layout \end_inset \begin_inset Text \begin_layout Standard 0.667 \end_layout \end_inset \begin_inset Text \begin_layout Standard e \end_layout \end_inset \begin_inset Text \begin_layout Standard floating point exponential \end_layout \end_inset \begin_inset Text \begin_layout Standard '%6.2e'%(2.0/3) \end_layout \end_inset \begin_inset Text \begin_layout Standard 6.67e-01 \end_layout \end_inset \begin_inset Text \begin_layout Standard x \end_layout \end_inset \begin_inset Text \begin_layout Standard hexadecimal \end_layout \end_inset \begin_inset Text \begin_layout Standard '%x'%(16) \end_layout \end_inset \begin_inset Text \begin_layout Standard 10 \end_layout \end_inset \begin_inset Text \begin_layout Standard o \end_layout \end_inset \begin_inset Text \begin_layout Standard octal \end_layout \end_inset \begin_inset Text \begin_layout Standard '%o'%(8) \end_layout \end_inset \begin_inset Text \begin_layout Standard 10 \end_layout \end_inset \begin_inset Text \begin_layout Standard s \end_layout \end_inset \begin_inset Text \begin_layout Standard string \end_layout \end_inset \begin_inset Text \begin_layout Standard '%s'%('abcd') \end_layout \end_inset \begin_inset Text \begin_layout Standard abcd \end_layout \end_inset \begin_inset Text \begin_layout Standard 0d \end_layout \end_inset \begin_inset Text \begin_layout Standard modified 'd' \end_layout \end_inset \begin_inset Text \begin_layout Standard '%05d'%(12) \end_layout \end_inset \begin_inset Text \begin_layout Standard 00012 \end_layout \end_inset \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Formatted Printing in Python \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left The following example shows some of the features available with formatted printing. \end_layout \begin_layout Standard \align left \emph on Example: format2.py \end_layout \begin_layout LyX-Code a = 'justify as you like' \end_layout \begin_layout LyX-Code print '%30s'%a # right justified \end_layout \begin_layout LyX-Code print '%-30s'%a # minus sign for left justification \end_layout \begin_layout LyX-Code for k in range(1,11): # A good looking table \end_layout \begin_layout LyX-Code print '5 x %2d = %2d' %(k, k*5) \end_layout \begin_layout Standard The output of \emph on format2.py \emph default is given below. \end_layout \begin_layout LyX-Code justify as you like \end_layout \begin_layout LyX-Code justify as you like \end_layout \begin_layout LyX-Code 5 x 1 = 5 \end_layout \begin_layout LyX-Code 5 x 2 = 10 \end_layout \begin_layout LyX-Code 5 x 3 = 15 \end_layout \begin_layout LyX-Code 5 x 4 = 20 \end_layout \begin_layout LyX-Code 5 x 5 = 25 \end_layout \begin_layout LyX-Code 5 x 6 = 30 \end_layout \begin_layout LyX-Code 5 x 7 = 35 \end_layout \begin_layout LyX-Code 5 x 8 = 40 \end_layout \begin_layout LyX-Code 5 x 9 = 45 \end_layout \begin_layout LyX-Code 5 x 10 = 50 \end_layout \begin_layout Section Exception Handling \end_layout \begin_layout Standard Errors detected during execution are called exceptions, like divide by zero. If the program does not handle exceptions, the Python Interpreter reports the exception and terminates the program. We will demonstrate handling exceptions using \shape italic try \shape default and \shape italic except \shape default keywords, in the example except.py. \end_layout \begin_layout Standard \align left \emph on Example: except.py \end_layout \begin_layout LyX-Code x = input('Enter a number ') \end_layout \begin_layout LyX-Code try: \end_layout \begin_layout LyX-Code print 10.0/x \end_layout \begin_layout LyX-Code except: \end_layout \begin_layout LyX-Code print 'Division by zero not allowed' \end_layout \begin_layout Standard If any exception occurs while running the code inside the try block, the code inside the except block is executed. The following program implements error checking on input using exceptions. \end_layout \begin_layout Standard \align left \emph on Example: except2.py \end_layout \begin_layout LyX-Code def get_number(): \end_layout \begin_layout LyX-Code while 1: \end_layout \begin_layout LyX-Code try: \end_layout \begin_layout LyX-Code a = raw_input('Enter a number ') \end_layout \begin_layout LyX-Code x = atof(a) \end_layout \begin_layout LyX-Code return x \end_layout \begin_layout LyX-Code except: \end_layout \begin_layout LyX-Code print 'Enter a valid number' \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print get_number() \end_layout \begin_layout Section Turtle Graphics \end_layout \begin_layout Standard Turtle Graphics have been noted by many psychologists and educators to be a powerful aid in teaching geometry, spatial perception, logic skills, computer programming, and art. The language LOGO was specifically designed to introduce children to programmin g, using turtle graphics. An abstract drawing device, called the Turtle, is used to make programming attractive for children by concentrating on doing turtle graphics. It has been used with children as young as 3 and has a track record of 30 years of success in education. \end_layout \begin_layout Standard We will use the Turtle module of Python to play with Turtle Graphics and practice the logic required for writing computer programs. Using this module, we will move a \shape italic Pen \shape default on a two dimensional screen to generate graphical patterns. The Pen can be controlled using functions like forward(distance), backward(dist ance), right(angle), left(angle) etc. \begin_inset Foot status collapsed \begin_layout Standard http://docs.python.org/library/turtle.html \end_layout \end_inset . Run the program turtle1.py to understand the functions. This section is included only for those who want to practice programming in a more interesting manner. \end_layout \begin_layout Standard \align left \emph on Example turtle1.py \end_layout \begin_layout LyX-Code from turtle import * \end_layout \begin_layout LyX-Code a = Pen() # Creates a turtle in a window \end_layout \begin_layout LyX-Code a.forward(50) \end_layout \begin_layout LyX-Code a.left(45) \end_layout \begin_layout LyX-Code a.backward(50) \end_layout \begin_layout LyX-Code a.right(45) \end_layout \begin_layout LyX-Code a.forward(50) \end_layout \begin_layout LyX-Code a.circle(10) \end_layout \begin_layout LyX-Code a.up() \end_layout \begin_layout LyX-Code a.forward(50) \end_layout \begin_layout LyX-Code a.down() \end_layout \begin_layout LyX-Code a.color('red') \end_layout \begin_layout LyX-Code a.right(90) \end_layout \begin_layout LyX-Code a.forward(50) \end_layout \begin_layout LyX-Code raw_input('Press Enter') \end_layout \begin_layout Standard \align left \emph on Example turtle2.py \end_layout \begin_layout LyX-Code from turtle import * \end_layout \begin_layout LyX-Code a = Pen() \end_layout \begin_layout LyX-Code for k in range(4): \end_layout \begin_layout LyX-Code a.forward(50) \end_layout \begin_layout LyX-Code a.left(90) \end_layout \begin_layout LyX-Code a.circle(25) \end_layout \begin_layout LyX-Code raw_input() # Wait for Key press \end_layout \begin_layout Standard Outputs of the program turtle2.py and turtle3.py are shown in figure \begin_inset LatexCommand ref reference "fig:turtle2 and 3 outputs" \end_inset . Try to write more programs like this to generate more complex patterns. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/turtle2.png width 3cm \end_inset \begin_inset Graphics filename pics/turtle3.png width 3cm \end_inset \begin_inset Graphics filename pics/turtle4.png width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of turtle2.py (b) turtle3.py (c) turtle4.py \begin_inset LatexCommand label name "fig:turtle2 and 3 outputs" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example turtle3.py \end_layout \begin_layout LyX-Code from turtle import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def draw_rectangle(): \end_layout \begin_layout LyX-Code for k in range(4): \end_layout \begin_layout LyX-Code a.forward(50) \end_layout \begin_layout LyX-Code a.left(90) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code a = Pen() \end_layout \begin_layout LyX-Code for k in range(36): \end_layout \begin_layout LyX-Code draw_rectangle() \end_layout \begin_layout LyX-Code a.left(10) \end_layout \begin_layout LyX-Code raw_input() \end_layout \begin_layout Standard The program turtle3.py creates a pattern by drwaing 36 squares, each drawn tilted by \begin_inset Formula $10^{\circ}$ \end_inset from the previous one. The program turtle4.py generates the fractal image as shown in figure \begin_inset LatexCommand ref reference "fig:turtle2 and 3 outputs" \end_inset (c). \end_layout \begin_layout Standard \align left \emph on Example turtle4.py \end_layout \begin_layout LyX-Code from turtle import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def f(length, depth): \end_layout \begin_layout LyX-Code if depth == 0: \end_layout \begin_layout LyX-Code forward(length) \end_layout \begin_layout LyX-Code else: \end_layout \begin_layout LyX-Code f(length/3, depth-1) \end_layout \begin_layout LyX-Code right(60) \end_layout \begin_layout LyX-Code f(length/3, depth-1) \end_layout \begin_layout LyX-Code left(120) \end_layout \begin_layout LyX-Code f(length/3, depth-1) \end_layout \begin_layout LyX-Code right(60) \end_layout \begin_layout LyX-Code f(length/3, depth-1) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code f(500, 4) \end_layout \begin_layout LyX-Code raw_input('Press any Key') \end_layout \begin_layout Section Writing GUI Programs \end_layout \begin_layout Standard Python has several modules that can be used for creating Graphical User Interfaces. The intention of this chapter is just to show the ease of making GUI in Python and we have selected Tkinter \begin_inset Foot status collapsed \begin_layout Standard http://www.pythonware.com/library/an-introduction-to-tkinter.htm \end_layout \begin_layout Standard http://infohost.nmt.edu/tcc/help/pubs/tkinter/ \end_layout \begin_layout Standard http://wiki.python.org/moin/TkInter \end_layout \end_inset , one of the easiest to learn. The GUI programs are event driven (movement of mouse, clicking a mouse button, pressing and releasing a key on the keyboard etc. are called events). The execution sequence of the program is decided by the events, generated mostly by the user. For example, when the user clicks on a Button, the code associated with that Button is executed. GUI Programming is about creating Widgets like Button, Label, Canvas etc. on the screen and executing selected functions in response to events. After creating all the necessary widgets and displaying them on the screen, the control is passed on to Tkinter by calling a function named \shape italic mainloop \shape default . After that the program flow is decided by the events and associated callback functions. \end_layout \begin_layout Standard For writing GUI programs, the first step is to create a main graphics window by calling the function Tk(). After that we create various Widgets and pack them inside the main window. The example programs given below demonstrate the usage of some of the Tkinter widgets.The program \shape italic tkmain.py \shape default is the smallest GUI program one can write using Tkinter. The output of tkmain.py is shown in figure \begin_inset LatexCommand ref reference "fig:tkmain and tklabel outputs" \end_inset (a). \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Standard \align center \begin_inset Graphics filename pics/tkmain.png width 4cm \end_inset \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset Graphics filename pics/tklabel.png width 4cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Outputs of (a)tkmain.py (b)tklabel.py \begin_inset LatexCommand label name "fig:tkmain and tklabel outputs" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example tkmain.py \end_layout \begin_layout LyX-Code from Tkinter import * \end_layout \begin_layout LyX-Code root = Tk() \end_layout \begin_layout LyX-Code root.mainloop() \end_layout \begin_layout Standard \align left \emph on Example tklabel.py \end_layout \begin_layout LyX-Code from Tkinter import * \end_layout \begin_layout LyX-Code root = Tk() \end_layout \begin_layout LyX-Code w = Label(root, text="Hello, world") \end_layout \begin_layout LyX-Code w.pack() \end_layout \begin_layout LyX-Code root.mainloop() \end_layout \begin_layout Standard The program tklabel.py will generate the output as shown in figure \begin_inset LatexCommand ref reference "fig:tkmain and tklabel outputs" \end_inset (b). Terminate the program by clicking on the x displayed at the top right corner. In this example, we used a Label widget to display some text. The next example will show how to use a Button widget. \end_layout \begin_layout Standard A Button widget can have a callback function, hello() in this case, that gets executed when the user clicks on the Button. The program will display a Button on the screen. Every time you click on it, the function \shape italic hello \shape default will be executed. The output of the program is shown in figure \begin_inset LatexCommand ref reference "fig:tkbutton and canvas" \end_inset (a). \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/tkbutton.png width 4cm \end_inset \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset Graphics filename pics/tkcanvas.png width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Outputs of (a) tkbutton.py (b)tkcanvas.py \begin_inset LatexCommand label name "fig:tkbutton and canvas" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example tkbutton.py \end_layout \begin_layout LyX-Code from Tkinter import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def hello(): \end_layout \begin_layout LyX-Code print 'hello world' \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code w = Tk() # Creates the main Graphics window \end_layout \begin_layout LyX-Code b = Button(w, text = 'Click Me', command = hello) \end_layout \begin_layout LyX-Code b.pack() \end_layout \begin_layout LyX-Code w.mainloop() \end_layout \begin_layout Standard Canvas is another commonly used widget. Canvas is a drawing area on which we can draw elements like line, arc, rectangle, text etc. The program tkcanvas.py creates a Canvas widget and binds the event to the function draw(). When left mouse button is pressed, a small rectangle are drawn at the cursor position. The output of the program is shown in figure \begin_inset LatexCommand ref reference "fig:tkbutton and canvas" \end_inset (b). \end_layout \begin_layout Standard \align left \emph on Example tkcanvas.py \end_layout \begin_layout LyX-Code from Tkinter import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def draw(event): \end_layout \begin_layout LyX-Code c.create_rectangle(event.x, \backslash \end_layout \begin_layout LyX-Code event.y, event.x+5, event.y+5) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code w = Tk() \end_layout \begin_layout LyX-Code c = Canvas(w, width = 300, height = 200) \end_layout \begin_layout LyX-Code c.pack() \end_layout \begin_layout LyX-Code c.bind("", draw) \end_layout \begin_layout LyX-Code w.mainloop() \end_layout \begin_layout Standard The next program is a modification of tkcanvas.py. The right mouse-button is bound to remove(). Every time a rectangle is drawn, its return value is added to a list, a global variable, and this list is used for removing the rectangles when right button is pressed. \end_layout \begin_layout Standard \align left \emph on Example tkcanvas2.py \end_layout \begin_layout LyX-Code from Tkinter import * \end_layout \begin_layout LyX-Code recs = [] # List keeping track of the rectangles \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def remove(event): \end_layout \begin_layout LyX-Code global recs \end_layout \begin_layout LyX-Code if len(recs) > 0: \end_layout \begin_layout LyX-Code c.delete(recs[0]) # delete from Canvas \end_layout \begin_layout LyX-Code recs.pop(0) # delete first item from list \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def draw(event): \end_layout \begin_layout LyX-Code global recs \end_layout \begin_layout LyX-Code r = c.create_rectangle(event.x, \backslash \end_layout \begin_layout LyX-Code event.y, event.x+5, event.y+5) \end_layout \begin_layout LyX-Code recs.append(r) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code w = Tk() \end_layout \begin_layout LyX-Code c = Canvas(w, width = 300, height = 200) \end_layout \begin_layout LyX-Code c.pack() \end_layout \begin_layout LyX-Code c.bind("", draw) \end_layout \begin_layout LyX-Code c.bind("", remove) \end_layout \begin_layout LyX-Code w.mainloop() \end_layout \begin_layout Section Object Oriented Programming in Python \end_layout \begin_layout Standard OOP is a programming paradigm that uses \emph on objects \emph default (Structures consisting of variables and methods) and their interactions to design computer programs. Python is an object oriented language but it does not force you to make all programs object oriented and there is no advantage in making small programs object oriented. In this section, we will discuss some features of OOP. \end_layout \begin_layout Standard Before going to the new concepts, let us recollect some of the things we have learned. We have seen that the effect of operators on different data types is predefined. For example \begin_inset Formula $2*3$ \end_inset results in \begin_inset Formula $6$ \end_inset and \begin_inset Formula $2*'abc'$ \end_inset results in \begin_inset Formula $'abcabc'$ \end_inset . This behavior has been decided beforehand, based on some logic, by the language designers. One of the key features of OOP is the ability to create user defined data types. The user will specify, how the new data type will behave under the existing operators like add, subtract etc. and also define methods that will belong to the new data type. \end_layout \begin_layout Standard We will design a new data type using the class keyword and define the behavior of it. In the program point.py, we define a class named Point. The variables xpos and ypos are members of Point. The __init__() function is executed whenever we create an instance of this class, the member variables are initialized by this function. The way in which an object belonging to this class is printed is decided by the __str__ function. We also have defined the behavior of add (+) and subtract (-) operators for this class. The + operator returns a new Point by adding the x and y coordinates of two Points. Subtracting a Point from another gives the distance between the two. The method dist() returns the distance of a Point object from the origin. We have not defined the behavior of Point under copy operation. We can use the copy module of Python to copy objects. \end_layout \begin_layout Standard \align left \emph on Example point.py \end_layout \begin_layout LyX-Code class Point: \end_layout \begin_layout LyX-Code ''' \end_layout \begin_layout LyX-Code This is documentation comment. \end_layout \begin_layout LyX-Code help(Point) will display this. \end_layout \begin_layout LyX-Code ''' \end_layout \begin_layout LyX-Code def __init__(self, x=0, y=0): \end_layout \begin_layout LyX-Code self.xpos = x \end_layout \begin_layout LyX-Code self.ypos = y \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def __str__(self): # overloads print \end_layout \begin_layout LyX-Code return 'Point at (%f,%f)'%(self.xpos, self.ypos) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def __add__(self, other): #overloads + \end_layout \begin_layout LyX-Code xpos = self.xpos + other.xpos \end_layout \begin_layout LyX-Code ypos = self.ypos + other.ypos \end_layout \begin_layout LyX-Code return Point(xpos,ypos) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def __sub__(self, other): #overloads - \end_layout \begin_layout LyX-Code import math \end_layout \begin_layout LyX-Code dx = self.xpos - other.xpos \end_layout \begin_layout LyX-Code dy = self.ypos - other.ypos \end_layout \begin_layout LyX-Code return math.sqrt(dx**2+dy**2) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def dist(self): \end_layout \begin_layout LyX-Code import math \end_layout \begin_layout LyX-Code return math.sqrt(self.xpos**2 + self.ypos**2) \end_layout \begin_layout Standard The program point1.py imports the file point.py to use the class Point defined inside it to demonstrate the properties of the class. A self. is prefixed when a method refers to member of the same object. It refers to the variable used for invoking the method. \end_layout \begin_layout Standard \align left \emph on Example point1.py \end_layout \begin_layout LyX-Code from point import * \end_layout \begin_layout LyX-Code origin = Point() \end_layout \begin_layout LyX-Code print origin \end_layout \begin_layout LyX-Code p1 = Point(4,4) \end_layout \begin_layout LyX-Code p2 = Point(8,7) \end_layout \begin_layout LyX-Code print p1 \end_layout \begin_layout LyX-Code print p2 \end_layout \begin_layout LyX-Code print p1 + p2 \end_layout \begin_layout LyX-Code print p1 - p2 \end_layout \begin_layout LyX-Code print p1.dist() \end_layout \begin_layout Standard Output of program point1.py is shown below. \end_layout \begin_layout Quotation Point at (0.000000,0.000000) \end_layout \begin_layout Quotation Point at (4.000000,4.000000) \end_layout \begin_layout Quotation Point at (8.000000,7.000000) \end_layout \begin_layout Quotation Point at (12.000000,11.000000) \end_layout \begin_layout Quotation 5.0 \end_layout \begin_layout Quotation 5.65685424949 \end_layout \begin_layout Standard In this section, we have demonstrated the OO concepts like class, object and operator overloading. \end_layout \begin_layout Subsection Inheritance, reusing code \end_layout \begin_layout Standard \align left Reuse of code is one of the main advantages of object oriented programming. We can define another class that inherits all the properties of the Point class, as shown below. The __init__ function of colPoint calls the __init__ function of Point, to get all work except initilization of color done. All other methods and operator overloading defined for Point is inherited by colPoint. \end_layout \begin_layout Standard \align left \emph on Example cpoint.py \end_layout \begin_layout LyX-Code class colPoint(Point): #colPoint inherits Point \end_layout \begin_layout LyX-Code color = 'black' \end_layout \begin_layout LyX-Code def __init__(self,x=0,y=0,col='black'): \end_layout \begin_layout LyX-Code Point.__init__(self,x,y) \end_layout \begin_layout LyX-Code self.color = col \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def __str__(self): \end_layout \begin_layout LyX-Code return '%s colored Point at (%f,%f)'% \backslash \end_layout \begin_layout LyX-Code (self.color,self.xpos, self.ypos) \end_layout \begin_layout Standard \align left \emph on Example point2.py \end_layout \begin_layout LyX-Code from cpoint import * \end_layout \begin_layout LyX-Code p1 = Point(5,5) \end_layout \begin_layout LyX-Code rp1 = colPoint(2,2,'red') \end_layout \begin_layout LyX-Code print p1 \end_layout \begin_layout LyX-Code print rp1 \end_layout \begin_layout LyX-Code print rp1 + p1 \end_layout \begin_layout LyX-Code print rp1.dist() \end_layout \begin_layout Standard The output of point2.py is listed below. \end_layout \begin_layout Standard Point at (5.000000,5.000000) \end_layout \begin_layout Standard red colored Point at (2.000000,2.000000) \end_layout \begin_layout Standard Point at (7.000000,7.000000) \end_layout \begin_layout Standard 2.82842712475 \end_layout \begin_layout Standard \align left For a detailed explanation on the object oriented features of Python, refer to chapters 13, 14 and 15 of the online book http://openbookproject.net//thinkCS py/ \end_layout \begin_layout Subsection A graphics example program \end_layout \begin_layout Standard Object Oriented programming allows us to write Classes with a well defined external interface hiding all the internal details. This example shows a Class named 'disp', for drawing curves, providing the xy coordinates within an arbitrary range . The the world-to-screen coordinate conversion is performed internally. The method named line() accepts a list of xy coordinates. The file \shape italic tkplot_class.py \shape default defines the 'disp' class and is listed below. \end_layout \begin_layout Standard \align left \emph on Example tkplot_class.py \end_layout \begin_layout LyX-Code from Tkinter import * \end_layout \begin_layout LyX-Code from math import * \end_layout \begin_layout LyX-Code class disp: \end_layout \begin_layout LyX-Code def __init__(self, parent, width=400., height=200.): \end_layout \begin_layout LyX-Code self.parent = parent \end_layout \begin_layout LyX-Code self.SCX = width \end_layout \begin_layout LyX-Code self.SCY = height \end_layout \begin_layout LyX-Code self.border = 1 \end_layout \begin_layout LyX-Code self.canvas = Canvas(parent, width=width, height=height) \end_layout \begin_layout LyX-Code self.canvas.pack(side = LEFT) \end_layout \begin_layout LyX-Code self.setWorld(0 , 0, self.SCX, self.SCY) # scale factors \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def setWorld(self, x1, y1, x2, y2): \end_layout \begin_layout LyX-Code self.xmin = float(x1) \end_layout \begin_layout LyX-Code self.ymin = float(y1) \end_layout \begin_layout LyX-Code self.xmax = float(x2) \end_layout \begin_layout LyX-Code self.ymax = float(y2) \end_layout \begin_layout LyX-Code self.xscale = (self.xmax - self.xmin) / (self.SCX) \end_layout \begin_layout LyX-Code self.yscale = (self.ymax - self.ymin) / (self.SCY) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def w2s(self, p): #world-to-screen before plotting \end_layout \begin_layout LyX-Code ip = [] \end_layout \begin_layout LyX-Code for xy in p: \end_layout \begin_layout LyX-Code ix = self.border + int( (xy[0] - self.xmin) / self.xscale) \end_layout \begin_layout LyX-Code iy = self.border + int( (xy[1] - self.ymin) / self.yscale) \end_layout \begin_layout LyX-Code iy = self.SCY - iy \end_layout \begin_layout LyX-Code ip.append((ix,iy)) \end_layout \begin_layout LyX-Code return ip \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def line(self, points, col='blue'): \end_layout \begin_layout LyX-Code ip = self.w2s(points) \end_layout \begin_layout LyX-Code t = self.canvas.create_line(ip, fill=col) \end_layout \begin_layout Standard The program \shape italic tkplot.py \shape default imports tkplot_class.py and plots two graphs. The advantage of code reuse is evident from this example. \begin_inset Foot status collapsed \begin_layout Standard A more sophisticated version of the \shape italic disp \shape default class program (draw.py) is included in the package 'learn-by-coding', available on the CD. \end_layout \end_inset . Output of \shape italic tkplot.py \shape default is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-tkplot.py" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/tkplot.png width 10cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of tkplot.py \begin_inset LatexCommand label name "fig:Output-of-tkplot.py" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example tkplot.py \end_layout \begin_layout LyX-Code from tkplot_class import * \end_layout \begin_layout LyX-Code from math import * \end_layout \begin_layout LyX-Code w = Tk() \end_layout \begin_layout LyX-Code gw1 = disp(w) \end_layout \begin_layout LyX-Code xy = [] \end_layout \begin_layout LyX-Code for k in range(200): \end_layout \begin_layout LyX-Code x = 2 * pi * k/200 \end_layout \begin_layout LyX-Code y = sin(x) \end_layout \begin_layout LyX-Code xy.append((x,y)) \end_layout \begin_layout LyX-Code gw1.setWorld(0, -1.0, 2*pi, 1.0) \end_layout \begin_layout LyX-Code gw1.line(xy) \end_layout \begin_layout LyX-Code gw2 = disp(w) \end_layout \begin_layout LyX-Code gw2.line([(10,10),(100,100),(350,50)], 'red') \end_layout \begin_layout LyX-Code w.mainloop() \end_layout \begin_layout Section Exercises \end_layout \begin_layout Enumerate Generate multiplication table of eight and write it to a file. \end_layout \begin_layout Enumerate Make a list and write it to a file using the pickle module. \end_layout \begin_layout Enumerate Write a Python program to open a file and write 'hello world' to it. \end_layout \begin_layout Enumerate Write a Python program to open a text file and read all lines from it. \end_layout \begin_layout Enumerate Write a program to generate the multiplication table of a number from the user. The output should be formatted as shown below \newline \InsetSpace ~ \InsetSpace ~ 1 x 5 = \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ 5 \newline \InsetSpace ~ \InsetSpace ~ 2 x 5 = \InsetSpace ~ 10 \end_layout \begin_layout Enumerate Define the list [1,2,3,4,5,6] using the range function. Write code to insert a 10 after 2, delete 4, add 0 at the end and sort it in the ascending order. \end_layout \begin_layout Enumerate Write Python code to generate the sequence of numbers \newline 25 20 15 10 5 \newline using range function . Delete 15 from the result and sort it. Print it using a for loop. \end_layout \begin_layout Enumerate Define a string \shape italic s = 'mary had a little lamb'. \newline \shape default a) print it in reverse order \newline b) split it using space character as sepatator \end_layout \begin_layout Enumerate Join the elements of the list ['I', 'am', 'in', 'pieces'] using + character. Do the same using a for loop also. \end_layout \begin_layout Enumerate Create a window with five buttons. Make each button a different color. Each button should have some text on it. \end_layout \begin_layout Enumerate Create a program that will put words in alphabetical order. The program should allow the user to enter as many words as he wants to. \end_layout \begin_layout Enumerate Create a program that will check a sentence to see if it is a palindrome. A palindrome is a sentence that reads the same backwards and forwards ('malayal am'). \end_layout \begin_layout Enumerate A text file contains two columns of numbers. Write a program to read them and print the sum of numbers in each row. \end_layout \begin_layout Enumerate Read a String from the keyboard. Multiply it by an integer to make its length more than 50. How do you find out the smallest number that does the job. \end_layout \begin_layout Enumerate Write a program to find the length of the hypotenuse of a right triangle from the length of other two sides, get the input from the user. \end_layout \begin_layout Enumerate Write a program displaying 2 labels and 2 buttons. It should print two different messages when clicked on the two buttons. \end_layout \begin_layout Enumerate Write a program with a Canvas and a circle drawn on it. \end_layout \begin_layout Enumerate Write a program using for loop to reverse a string. \end_layout \begin_layout Enumerate Write a Python function to calculate the GCD of two numbers \end_layout \begin_layout Enumerate Write a program to print the values of sine function from \begin_inset Formula $0$ \end_inset to \begin_inset Formula $2\pi$ \end_inset with 0.1 increments. Find the mean value of them. \end_layout \begin_layout Enumerate Generate N random numbers using random.random() and find out howmay are below 0.5 . Repeat the same for different values of N to draw some conclusions. \end_layout \begin_layout Enumerate Use the equation \begin_inset Formula $x=(-b\pm\sqrt{b^{2}-4ac})/2a$ \end_inset to find the roots of \begin_inset Formula $3x^{2}+6x+12=0$ \end_inset \end_layout \begin_layout Enumerate Write a program to calculate the distance between points (x1,y1) and (x2,y2) in a Cartesian plane. Get the coordinates from the user. \end_layout \begin_layout Enumerate Write a program to evaluate \begin_inset Formula $y$ \end_inset = \begin_inset Formula $\sqrt{2.3a}+a^{2}+34.5$ \end_inset for a = 1, 2 and 3. \end_layout \begin_layout Enumerate Print Fibanocci numbers upto 100, without using multiple assignment statement. \end_layout \begin_layout Enumerate Draw a chess board pattern using turtle graphics. \end_layout \begin_layout Enumerate Find the syntax error in the following code and correct it. \newline x=1 \newline while x <= 10: \newline print x * 5 \end_layout \begin_layout Chapter Arrays and Matrices \begin_inset LatexCommand label name "sec:Arrays-and-Matrices" \end_inset \end_layout \begin_layout Standard In the previous chapter, we have learned the essential features of Python language. We also used the \shape italic math \shape default module to calculate trigonometric functions. Using the tools introduced so far, let us generate the data points to plot a sine wave. The program sine.py generates the coordinates to plot a sine wave. \end_layout \begin_layout Standard \align left \emph on Example sine.py \end_layout \begin_layout LyX-Code import math \end_layout \begin_layout LyX-Code x = 0.0 \end_layout \begin_layout LyX-Code while x < 2 * math.pi: \end_layout \begin_layout LyX-Code print x , math.sin(x) \end_layout \begin_layout LyX-Code x = x + 0.1 \end_layout \begin_layout Standard The output to the screen can be redirected to a file as shown below, from the command prompt. You can plot the data using some program like xmgrace. \end_layout \begin_layout Standard $ python sine.py > sine.dat \end_layout \begin_layout Standard $ xmgrace sine.dat \end_layout \begin_layout Standard It would be better if we could write such programs without using loops explicitl y. Serious scientific computing requires manipulating of large data structures like matrices. The \shape italic list \shape default data type of Python is very flexible but the performance is not acceptable for large scale computing. The need of special tools is evident even from the simple example shown above. \shape italic NumPy \shape default is a package widely used for scientific computing with Python. \begin_inset Foot status collapsed \begin_layout Standard http://numpy.scipy.org/ \end_layout \begin_layout Standard http://www.scipy.org/Tentative_NumPy_Tutorial \end_layout \begin_layout Standard http://www.scipy.org/Numpy_Functions_by_Category \end_layout \begin_layout Standard http://www.scipy.org/Numpy_Example_List_With_Doc \end_layout \end_inset \end_layout \begin_layout Section The NumPy Module \end_layout \begin_layout Standard The \shape italic \color black numpy \shape default \color inherit module supports operations on compound data types like arrays and matrices. \shape italic \color black First thing to learn is how to create arrays and matrices using the numpy package. \shape default \color inherit Python lists can be converted into multi-dimensional arrays. There are several other functions that can be used for creating matrices. The mathematical functions like sine, cosine etc. of numpy accepts array objects as arguments and return the results as arrays objects. NumPy arrays can be indexed, sliced and copied like Python Lists. \end_layout \begin_layout Standard In the examples below, we will import numpy functions as local (using the syntax \shape italic from numpy import * \shape default ). Since it is the only package used there is no possibility of any function name conflicts. \end_layout \begin_layout Standard \align left \emph on Example numpy1.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code x = array( [1, 2, 3] ) # Make array from list \end_layout \begin_layout LyX-Code print x , type(x) \end_layout \begin_layout Standard In the above example, we have created an array from a list. \end_layout \begin_layout Subsection Creating Arrays and Matrices \end_layout \begin_layout Standard We can also make multi-dimensional arrays. Remember that a member of a list can be another list. The following example shows how to make a two dimensional array. \end_layout \begin_layout Standard \align left \emph on Example numpy3.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code a = [ [1,2] , [3,4] ] # make a list of lists \end_layout \begin_layout LyX-Code x = array(a) # and convert to an array \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout Standard Other than than \shape italic array(), \shape default there are several other functions that can be used for creating different types of arrays and matrices. Some of them are described below. \end_layout \begin_layout Subsubsection arange(start, stop, step, dtype = None) \end_layout \begin_layout Standard Creates an evenly spaced one-dimensional array. Start, stop, stepsize and datatype are the arguments. If datatype is not given, it is deduced from the other arguments. Note that, the values are generated within the interval, including start but excluding stop. \end_layout \begin_layout Standard arange(2.0, 3.0, .1) makes the array([ 2. , 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9]) \end_layout \begin_layout Subsubsection linspace(start, stop, number of elements) \end_layout \begin_layout Standard Similar to arange(). Start, stop and number of samples are the arguments. \end_layout \begin_layout Standard linspace(1, 2, 11) is equivalent to array([ 1. , 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2. ]) \end_layout \begin_layout Subsubsection zeros(shape, datatype) \end_layout \begin_layout Standard Returns a new array of given shape and type, filled zeros. The arguments are shape and datatype. For example \emph on \color black zeros( [3,2], 'float') \emph default \color inherit generates a 3 x 2 array filled with zeros as shown below. If not specified, the type of elements defaults to int. \end_layout \begin_layout Standard \begin_inset Formula $\begin{array}{ccc} 0.0 & 0.0 & 0.0\\ 0.0 & 0.0 & 0.0\end{array}$ \end_inset \end_layout \begin_layout Subsubsection ones(shape, datatype) \end_layout \begin_layout Standard Similar to zeros() except that the values are initialized to 1. \end_layout \begin_layout Subsubsection random.random(shape) \end_layout \begin_layout Standard Similar to the functions above, but the matrix is filled with random numbers ranging from 0 to 1, of \emph on \color black float \emph default \color inherit type. For example, random.random([3,3]) will generate the 3x3 matrix; \end_layout \begin_layout LyX-Code array([[ 0.3759652 , 0.58443562, 0.41632997], \end_layout \begin_layout LyX-Code [ 0.88497654, 0.79518478, 0.60402514], \end_layout \begin_layout LyX-Code [ 0.65468458, 0.05818105, 0.55621826]]) \end_layout \begin_layout Subsubsection reshape(array, newshape) \end_layout \begin_layout Standard We can also make multi-dimensions arrays by reshaping a one-dimensional array. The function \shape italic reshape() \shape default changes dimensions of an array. The total number of elements must be preserved. Working of \shape italic reshape() \shape default can be understood by looking at \emph on \color black reshape.py \emph default \color inherit and its result. \end_layout \begin_layout Standard \align left \emph on Example reshape.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code a = arange(20) \end_layout \begin_layout LyX-Code b = reshape(a, [4,5]) \end_layout \begin_layout LyX-Code print b \end_layout \begin_layout Standard The result is shown below. \end_layout \begin_layout LyX-Code array([[ 0, 1, 2, 3, 4], \end_layout \begin_layout LyX-Code [ 5, 6, 7, 8, 9], \end_layout \begin_layout LyX-Code [10, 11, 12, 13, 14], \end_layout \begin_layout LyX-Code [15, 16, 17, 18, 19]]) \end_layout \begin_layout Standard The program \shape italic numpy2.py \shape default demonstrates most of the functions discussed so far. \end_layout \begin_layout Standard \align left \emph on Example numpy2.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code a = arange(1.0, 2.0, 0.1) \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code b = linspace(1,2,11) \end_layout \begin_layout LyX-Code print b \end_layout \begin_layout LyX-Code c = ones(5,'float') \end_layout \begin_layout LyX-Code print c \end_layout \begin_layout LyX-Code d = zeros(5, 'int') \end_layout \begin_layout LyX-Code print d \end_layout \begin_layout LyX-Code e = random.rand(5) \end_layout \begin_layout LyX-Code print e \end_layout \begin_layout Standard Output of this program will look like; \end_layout \begin_layout Standard [ 1. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9] \end_layout \begin_layout Standard [ 1. 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2. ] \end_layout \begin_layout Standard [ 1. 1. 1. 1. 1.] \end_layout \begin_layout Standard [ 0. 0. 0. 0. 0.] \end_layout \begin_layout Standard [ 0.89039193 0.55640332 0.38962117 0.17238343 0.01297415] \end_layout \begin_layout Subsection Copying \end_layout \begin_layout Standard Numpy arrays can be copied using the copy method, as shown below. \end_layout \begin_layout Standard \align left \emph on Example array_copy.py \end_layout \begin_layout LyX-Code from mumpy import * \end_layout \begin_layout LyX-Code a = zeros(5) \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code b = a \end_layout \begin_layout LyX-Code c = a.copy() \end_layout \begin_layout LyX-Code c[0] = 10 \end_layout \begin_layout LyX-Code print a, c \end_layout \begin_layout LyX-Code b[0] = 10 \end_layout \begin_layout LyX-Code print a,c \end_layout \begin_layout Standard The output of the program is shown below. The statement \begin_inset Formula $b=a$ \end_inset does not make a copy of a. Modifying b affects a, but c is a separate entity. \end_layout \begin_layout Standard [ 0. 0. 0.] \end_layout \begin_layout Standard [ 0. 0. 0.] [ 10. 0. 0.] \end_layout \begin_layout Standard [ 10. 0. 0.] [ 10. 0. 0.] \end_layout \begin_layout Subsection Arithmetic Operations \end_layout \begin_layout Standard Arithmetic operations performed on an array is carried out on all individual elements. Adding or multiplying an array object with a number will multiply all the elements by that number. However, adding or multiplying two arrays having identical shapes will result in performing that operation with the corresponding elements. To clarify the idea, have a look at \emph on \color black aroper.py \emph default \color inherit and its results. \end_layout \begin_layout Standard \align left \emph on Example aroper.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code a = array([[2,3], [4,5]]) \end_layout \begin_layout LyX-Code b = array([[1,2], [3,0]]) \end_layout \begin_layout LyX-Code print a + b \end_layout \begin_layout LyX-Code print a * b \end_layout \begin_layout Standard The output will be as shown below \end_layout \begin_layout LyX-Code array([[3, 5], \end_layout \begin_layout LyX-Code [7, 5]]) \end_layout \begin_layout LyX-Code array([[ 2, 6], \end_layout \begin_layout LyX-Code [12, 0]]) \end_layout \begin_layout Standard Modifying this program for more operations is left as an exercise to the reader. \end_layout \begin_layout Subsection cross product \end_layout \begin_layout Standard Returns the cross product of two vectors, defined by \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} A\times B=\left|\begin{array}{clc} i & j & k\\ A_{1} & A_{2} & A_{3}\\ B_{1} & B_{2} & B_{3}\end{array}\right|=i(A_{2}B_{3}-A_{3}B_{2})+j(A_{1}B_{3}-A_{3}B_{1})+k(A_{1}B_{2}-A_{2}B_{1})\label{eq:cross product}\end{equation} \end_inset \end_layout \begin_layout Standard It can be evaluated using the function cross((array1, array2). The program \shape italic cross.py \shape default prints [-3, 6, -3] \end_layout \begin_layout Standard \align left \emph on Example cross.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code a = array([1,2,3]) \end_layout \begin_layout LyX-Code b = array([4,5,6]) \end_layout \begin_layout LyX-Code c = cross(a,b) \end_layout \begin_layout LyX-Code print c \end_layout \begin_layout Subsection dot product \end_layout \begin_layout Standard Returns the dot product of two vectors defined by \begin_inset Formula $A.B=A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}$ \end_inset . If you change the fourth line of \emph on \color black cross.py \emph default \color inherit to \begin_inset Formula $c=dot(a,b)$ \end_inset , the result will be 32. \end_layout \begin_layout Subsection Saving and Restoring \end_layout \begin_layout Standard An array can be saved to text file using \shape italic array.tofile(filename) \shape default and it can be read back using \shape italic array=fromfile() \shape default methods, as shown by the code fileio.py \end_layout \begin_layout Standard \align left \emph on Example fileio.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code a = arange(10) \end_layout \begin_layout LyX-Code a.tofile('myfile.dat') \end_layout \begin_layout LyX-Code b = fromfile('myfile.dat',dtype = 'int') \end_layout \begin_layout LyX-Code print b \end_layout \begin_layout Standard The function fromfile() sets dtype='float' by default. In this case we have saved an integer array and need to specify that while reading the file. We could have saved it as float the the statement a.tofile('myfile.dat', 'float'). \end_layout \begin_layout Subsection Matrix inversion \end_layout \begin_layout Standard The function \shape italic linalg.inv(matrix) \shape default computes the inverse of a square matrix, if it exists. We can verify the result by multiplying the original matrix with the inverse. Giving a singular matrix as the argument should normally result in an error message. In some cases, you may get a result whose elements are having very high values, and it indicates an error. \end_layout \begin_layout Standard \align left \emph on Example inv.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code a = array([ [4,1,-2], [2,-3,3], [-6,-2,1] ], dtype='float') \end_layout \begin_layout LyX-Code ainv = linalg.inv(a) \end_layout \begin_layout LyX-Code print ainv \end_layout \begin_layout LyX-Code print dot(a,ainv) \end_layout \begin_layout Standard Result of this program is printed below. \end_layout \begin_layout LyX-Code [[ 0.08333333 0.08333333 -0.08333333] \end_layout \begin_layout LyX-Code [-0.55555556 -0.22222222 -0.44444444] \end_layout \begin_layout LyX-Code [-0.61111111 0.05555556 -0.38888889]] \end_layout \begin_layout LyX-Code [[ 1.00000000e+00 -1.38777878e-17 0.00000000e+00] \end_layout \begin_layout LyX-Code [-1.11022302e-16 1.00000000e+00 0.00000000e+00] \end_layout \begin_layout LyX-Code [ 0.00000000e+00 2.08166817e-17 1.00000000e+00]] \end_layout \begin_layout Section Vectorized Functions \end_layout \begin_layout Standard The functions like sine, log etc. from NumPy are capable of accepting arrays as arguments. This eliminates the need of writing loops in our Python code. \end_layout \begin_layout Standard \align left \emph on Example vfunc.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code a = array([1,10,100,1000]) \end_layout \begin_layout LyX-Code print log10(a) \end_layout \begin_layout Standard The output of the program is [ 0. 1. 2. 3.] , where the log of each element is calculated and returned in an array. This feature simplifies the programs a lot. Numpy also provides a function to vectorize functions written by the user. \end_layout \begin_layout Standard \align left \emph on Example vectorize.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def spf(x): \end_layout \begin_layout LyX-Code return 3*x \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code vspf = vectorize(spf) \end_layout \begin_layout LyX-Code a = array([1,2,3,4]) \end_layout \begin_layout LyX-Code print vspf(a) \end_layout \begin_layout Standard The output will be [ 3 6 9 12] . \end_layout \begin_layout Section Exercises \end_layout \begin_layout Enumerate Write code to make a one dimensional matrix with elements 5,10,15,20 and 25. make another matrix by slicing the first three elements from it. \end_layout \begin_layout Enumerate Create a \begin_inset Formula $3\times2$ \end_inset matrix and print the sum of its elements using for loops. \end_layout \begin_layout Enumerate Create a \begin_inset Formula $2\times3$ \end_inset matrix and fill it with random numbers. \end_layout \begin_layout Enumerate Use linspace to make an array from 0 to 10, with stepsize of 0.1 \end_layout \begin_layout Enumerate Use arange to make an 100 element array ranging from 0 to 10 \end_layout \begin_layout Enumerate Make an array a = [2,3,4,5] and copy it to \shape italic b \shape default . change one element of \shape italic b \shape default and print both. \end_layout \begin_layout Enumerate Make a 3x3 matrix and multipy it by 5. \end_layout \begin_layout Enumerate Create two 3x3 matrices and add them. \end_layout \begin_layout Enumerate Write programs to demonstrate the dot and cross products. \end_layout \begin_layout Enumerate Using matrix inversion, solve the system of equations \newline \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ 4x1 − 2x2 + \InsetSpace ~ \InsetSpace ~ x3 = 11 \newline −2x1 + 4x2 − 2x3 = −16 \newline \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ x1 − 2x2 + 4x3 = 17 \end_layout \begin_layout Enumerate Find the new values of the coordinate (10,10) under a rotation by angle \begin_inset Formula $\pi/4$ \end_inset . \end_layout \begin_layout Enumerate Write a vectorized function to evaluate \begin_inset Formula $y=x^{2}$ \end_inset and print the result for x=[1,2,3]. \end_layout \begin_layout Chapter Data visualization \end_layout \begin_layout Standard A graph or chart is used to present numerical data in visual form. A graph is one of the easiest ways to compare numbers. They should be used to make facts clearer and more understandable. Results of mathematical computations are often presented in graphical format. In this chapter, we will explore the Python modules used for generating two and three dimensional graphs of various types. \end_layout \begin_layout Section The Matplotlib Module \end_layout \begin_layout Standard Matplotlib is a python package that produces publication quality figures in a variety of hardcopy formats. It also provides many functions for matrix manipulation. You can generate plots, histograms, power spectra, bar charts, error-charts, scatter-plots, etc, with just a few lines of code and have full control of line styles, font properties, axes properties, etc. The data points to the plotting functions are supplied as Python lists or Numpy arrays. \end_layout \begin_layout Standard If you import matplotlib as \emph on \color black pylab, \emph default \color inherit the plotting functions from the submodules \emph on \color black pyplot \emph default \color inherit and matrix manipulation functions from the submodule \emph on \color black mlab \emph default \color inherit will be available as local functions. Pylab also imports Numpy for you. Let us start with some simple plots to become familiar with matplotlib. \begin_inset Foot status collapsed \begin_layout Standard http://matplotlib.sourceforge.net/ \end_layout \begin_layout Standard http://matplotlib.sourceforge.net/users/pyplot_tutorial.html \end_layout \begin_layout Standard http://matplotlib.sourceforge.net/examples/index.html \end_layout \begin_layout Standard http://matplotlib.sourceforge.net/api/axes_api.html \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example plot1.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code data = [1,2,5] \end_layout \begin_layout LyX-Code plot(data) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard In the above example, the x-axis of the three points is taken from 0 to 2. We can specify both the axes as shown below. \end_layout \begin_layout Standard \align left \emph on Example plot2.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code x = [1,2,5] \end_layout \begin_layout LyX-Code y = [4,5,6] \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard By default, the color is blue and the line style is continuous. This can be changed by an optional argument after the coordinate data, which is the format string that indicates the color and line type of the plot. The default format string is ‘b-‘ (blue, continuous line). Let us rewrite the above example to plot using red circles. We will also set the ranges for x and y axes and label them. \end_layout \begin_layout Standard \align left \emph on Example plot3.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code x = [1,2,5] \end_layout \begin_layout LyX-Code y = [4,5,6] \end_layout \begin_layout LyX-Code plot(x,y,'ro') \end_layout \begin_layout LyX-Code xlabel('x-axis') \end_layout \begin_layout LyX-Code ylabel('y-axis') \end_layout \begin_layout LyX-Code axis([0,6,1,7]) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/plot4.png lyxscale 40 width 4cm \end_inset \begin_inset Graphics filename pics/subplot1.png lyxscale 40 width 4cm \end_inset \begin_inset Graphics filename pics/piechart.png lyxscale 40 width 4cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of (a) plot4.py (b) subplot1.py (c) piechart.py \begin_inset LatexCommand label name "fig:Output-of-plot4.py" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The figure \begin_inset LatexCommand ref reference "fig:Output-of-plot4.py" \end_inset shows two different plots in the same window, using different markers and colors. \end_layout \begin_layout Standard \align left \emph on Example plot4.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code t = arange(0.0, 5.0, 0.2) \end_layout \begin_layout LyX-Code plot(t, t**2,'x') # \begin_inset Formula $t^{2}$ \end_inset \end_layout \begin_layout LyX-Code plot(t, t**3,'ro') # \begin_inset Formula $t^{3}$ \end_inset \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard We have just learned how to draw a simple plot using the pylab interface of matplotlib. \end_layout \begin_layout Subsection Multiple plots \end_layout \begin_layout Standard Matplotlib allows you to have multiple plots in the same window, using the subplot() command as shown in the example subplot1.py, whose output is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-plot4.py" \end_inset (b). \end_layout \begin_layout Standard \align left \emph on Example subplot1.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code subplot(2,1,1) # the first subplot \end_layout \begin_layout LyX-Code plot([1,2,3,4]) \end_layout \begin_layout LyX-Code subplot(2,1,2) # the second subplot \end_layout \begin_layout LyX-Code plot([4,2,3,1]) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard The arguments to subplot function are NR (number of rows) , NC (number of columns) and a figure number, that ranges from 1 to \begin_inset Formula $NR*NC$ \end_inset . The commas between the arguments are optional if \begin_inset Formula $NR*NC<10$ \end_inset , ie. subplot(2,1,1) can be written as subplot(211). \end_layout \begin_layout Standard Another example of subplot is given is \shape italic subplot2.py \shape default . You can modify the variable NR and NC to watch the results. Please note that the % character has different meanings. In \shape italic (pn+1)%5 \shape default , it is the reminder operator resulting in a number less than 5. The % character also appears in the String formatting. \end_layout \begin_layout Standard \align left \emph on Example subplot2.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code mark = ['x','o','^','+','>'] \end_layout \begin_layout LyX-Code NR = 2 # number of rows \end_layout \begin_layout LyX-Code NC = 3 # number of columns \end_layout \begin_layout LyX-Code pn = 1 \end_layout \begin_layout LyX-Code for row in range(NR): \end_layout \begin_layout LyX-Code for col in range(NC): \end_layout \begin_layout LyX-Code subplot(NR, NC, pn) \end_layout \begin_layout LyX-Code a = rand(10) * pn \end_layout \begin_layout LyX-Code plot(a, marker = mark[(pn+1)%5]) \end_layout \begin_layout LyX-Code xlabel('plot %d X'%pn) \end_layout \begin_layout LyX-Code ylabel('plot %d Y'%pn) \end_layout \begin_layout LyX-Code pn = pn + 1 \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Subsection Polar plots \end_layout \begin_layout Standard Polar coordinates locate a point on a plane with one distance and one angle. The distance ‘r’ is measured from the origin. The angle \begin_inset Formula $\theta$ \end_inset is measured from some agreed starting point. Use the positive part of the \begin_inset Formula $x-axis$ \end_inset as the starting point for measuring angles. Measure positive angles anti-clockwise from the positive \begin_inset Formula $x-axis$ \end_inset and negative angles clockwise from it. \end_layout \begin_layout Standard Matplotlib supports polar plots, using the polar( \begin_inset Formula $\theta,r$ \end_inset ) function. Let us plot a circle using polar(). For every point on the circle, the value of \begin_inset Formula $radius$ \end_inset is the same but the polar angle \begin_inset Formula $\theta$ \end_inset changes from \begin_inset Formula $0$ \end_inset to \begin_inset Formula $2\pi$ \end_inset . Both the coordinate arguments must be arrays of equal size. Since \begin_inset Formula $\theta$ \end_inset is having 100 points , \begin_inset Formula $r$ \end_inset also must have the same number. This array can be generated using the \begin_inset Formula $ones()$ \end_inset function. The axis([ \begin_inset Formula $\theta_{min},\theta_{max},r_{min},r_{max}$ \end_inset ) function can be used for setting the scale. \end_layout \begin_layout Standard \align left \emph on Example polar.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code th = linspace(0,2*pi,100) \end_layout \begin_layout LyX-Code r = 5 * ones(100) # radius = 5 \end_layout \begin_layout LyX-Code polar(th,r) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Subsection Pie Charts \end_layout \begin_layout Standard An example of a pie chart is given below. The percentage of different items and their names are given as arguments. The output is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-plot4.py" \end_inset (c). \end_layout \begin_layout Standard \align left \emph on Example piechart.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code labels = 'Frogs', 'Hogs', 'Dogs', 'Logs' \end_layout \begin_layout LyX-Code fracs = [25, 25, 30, 20] \end_layout \begin_layout LyX-Code pie(fracs, labels=labels) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section Plotting mathematical functions \end_layout \begin_layout Standard One of our objectives is to understand different mathematical functions better, by plotting them graphically. We will use the \shape italic arange \shape default , \shape italic linspace \shape default and \shape italic logspace \shape default functions from \shape italic numpy \shape default to generate the input data and also the vectorized versions of the mathematical functions. For arange(), the third argument is the stepsize. The total number of elements is calculated from start, stop and stepsize. In the case of linspace(), we provide start, stop and the total number of points. The step size is calculated from these three parameters. Please note that to create a data set ranging from 0 to 1 (including both) with a stepsize of 0.1, we need to specify linspace(0,1,11) and not linspace(0,1 ,10). \end_layout \begin_layout Subsection Sine function and friends \end_layout \begin_layout Standard Let the first example be the familiar sine function. The input data is from \begin_inset Formula $-\pi$ \end_inset to \begin_inset Formula $+\pi$ \end_inset radians \begin_inset Foot status collapsed \begin_layout Standard Why do we need to give the angles in radians and not in degrees. Angle in radian is the length of the arc defined by the given angle, with unit radius. Degree is just an arbitrary unit. \end_layout \end_inset . To make it a bit more interesting we are plotting \begin_inset Formula $\sin x^{2}$ \end_inset also. The objective is to explain the concept of odd and even functions. Mathematically, we say that a function \begin_inset Formula $f(x)$ \end_inset is even if \begin_inset Formula $f(x)=f(-x)$ \end_inset and is odd if \begin_inset Formula $f(-x)=-f(x)$ \end_inset . Even functions are functions for which the left half of the plane looks like the mirror image of the right half of the plane. From the figure \begin_inset LatexCommand ref reference "fig:Sine and Circ" \end_inset (a) you can see that \begin_inset Formula $\sin x$ \end_inset is odd and \begin_inset Formula $\sin x^{2}$ \end_inset is even. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/npsin.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/circ.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard (a) Output of npsin.py (b) Output of circ.py \begin_inset LatexCommand label name "fig:Sine and Circ" \end_inset . \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example npsin.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code x = linspace(-pi, pi , 200) \end_layout \begin_layout LyX-Code y = sin(x) \end_layout \begin_layout LyX-Code y1 = sin(x*x) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code plot(x,y1,'r') \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard Exercise: Modify the program \emph on \color black npsin.py \emph default \color inherit to plot \begin_inset Formula $\sin^{2}x$ \end_inset , \begin_inset Formula $\cos x$ \end_inset , \begin_inset Formula $\sin x^{3}$ \end_inset etc. \end_layout \begin_layout Subsection Trouble with Circle \end_layout \begin_layout Standard Equation of a circle is \begin_inset Formula $x^{2}+y^{2}=a^{2}$ \end_inset , where a is the radius and the circle is located at the origin of the coordinate system. In order to plot it using Cartesian coordinates, we need to express \begin_inset Formula $y$ \end_inset in terms of \begin_inset Formula $x$ \end_inset , and is given by \begin_inset Formula \[ y=\sqrt{a^{2}-x^{2}}\] \end_inset \end_layout \begin_layout Standard We will create the x-coordinates ranging from \begin_inset Formula $-a$ \end_inset to \begin_inset Formula $+a$ \end_inset and calculate the corresponding values of y. This will give us only half of the circle, since for each value of x, there are two values of y (+y and -y). The following program \emph on \color black circ.py \emph default \color inherit creates both to make the complete circle as shown in figure \begin_inset LatexCommand ref reference "fig:Sine and Circ" \end_inset (b). Any multi-valued function will have this problem while plotting. Such functions can be plotted better using parametric equations or using the polar plot options, as explained in the coming sections. \end_layout \begin_layout Standard \align left \emph on Example circ.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 10.0 \end_layout \begin_layout LyX-Code x = linspace(-a, a , 200) \end_layout \begin_layout LyX-Code yupper = sqrt(a**2 - x**2) \end_layout \begin_layout LyX-Code ylower = -sqrt(a**2 - x**2) \end_layout \begin_layout LyX-Code plot(x,yupper) \end_layout \begin_layout LyX-Code plot(x,ylower) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Subsection Parametric plots \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Standard \align center \begin_inset Graphics filename pics/circpar.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/arcs.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard (a)Output of circpar.py. (b)Output of arcs.py \begin_inset LatexCommand label name "fig:(a)Circpar and Arc" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard The circle can be represented using the equations \begin_inset Formula $x=a\cos\theta$ \end_inset and \begin_inset Formula $y=a\sin\theta$ \end_inset . To get the complete circle \begin_inset Formula $\theta$ \end_inset should vary from zero to \begin_inset Formula $2\pi$ \end_inset radians. The output of circpar.py is shown in figure \begin_inset LatexCommand ref reference "fig:(a)Circpar and Arc" \end_inset (a). \end_layout \begin_layout Standard \align left \emph on Example circpar.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 10.0 \end_layout \begin_layout LyX-Code th = linspace(0, 2*pi, 200) \end_layout \begin_layout LyX-Code x = a * cos(th) \end_layout \begin_layout LyX-Code y = a * sin(th) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard Changing the range of \begin_inset Formula $\theta$ \end_inset to less than \begin_inset Formula $2\pi$ \end_inset radians will result in an arc. The following example plots several arcs with different radii. The \emph on \color black for \emph default \color inherit loop will execute four times with the values of radius 5,10,15 and 20. The range of \begin_inset Formula $\theta$ \end_inset also depends on the loop variable. For the next three values it will be \begin_inset Formula $\pi,1.5\pi and2\pi$ \end_inset respectively. The output is shown in figure \begin_inset LatexCommand ref reference "fig:(a)Circpar and Arc" \end_inset (b). \end_layout \begin_layout Standard \align left \emph on Example arcs.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 10.0 \end_layout \begin_layout LyX-Code for a in range(5,21,5): \end_layout \begin_layout LyX-Code th = linspace(0, pi * a/10, 200) \end_layout \begin_layout LyX-Code x = a * cos(th) \end_layout \begin_layout LyX-Code y = a * sin(th) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section Famous Curves \end_layout \begin_layout Standard Connection between different branches of mathematics like trigonometry, algebra and geometry can be understood by geometrically representing the equations. You will find a large number of equations generating geometric patterns having interesting symmetries. A collection of them is available on the Internet \begin_inset LatexCommand cite key "wikipedia" \end_inset \begin_inset LatexCommand cite key "gap-system" \end_inset . We will select some of them and plot here. Exploring them further is left as an exercise to the reader. \end_layout \begin_layout Subsection Astroid \end_layout \begin_layout Standard The astroid was first discussed by Johann Bernoulli in 1691-92. It also appears in Leibniz's correspondence of 1715. It is sometimes called the tetracuspid for the obvious reason that it has four cusps. A circle of radius 1/4 rolls around inside a circle of radius 1 and a point on its circumference traces an astroid. The Cartesian equation is \begin_inset Formula \begin{equation} x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\label{eq:Astroid-cart}\end{equation} \end_inset \end_layout \begin_layout Standard The parametric equations are \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} x=a\cos^{3}(t),y=a\sin^{3}(t)\label{eq:Astroid-Par}\end{equation} \end_inset \end_layout \begin_layout Standard In order to plot the curve in the Cartesian system, we rewrite equation \begin_inset LatexCommand ref reference "eq:Astroid-cart" \end_inset as \end_layout \begin_layout Standard \begin_inset Formula \[ y=(a^{\frac{2}{3}}-x^{\frac{2}{3}})^{\frac{3}{2}}\] \end_inset \end_layout \begin_layout Standard The program \emph on \color black astro.py \emph default \color inherit plots the part of the curve in the first quadrant. The program \emph on \color black astropar.py \emph default \color inherit uses the parametric equation and plots the complete curve. Both are shown in figure \begin_inset LatexCommand ref reference "fig:(a)Astro.py" \end_inset \end_layout \begin_layout Standard \align left \emph on Example astro.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 2 \end_layout \begin_layout LyX-Code x = linspace(0,a,100) \end_layout \begin_layout LyX-Code y = ( a**(2.0/3) - x**(2.0/3) )**(3.0/2) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard \align left \emph on Example astropar.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 2 \end_layout \begin_layout LyX-Code t = linspace(-2*a,2*a,101) \end_layout \begin_layout LyX-Code x = a * cos(t)**3 \end_layout \begin_layout LyX-Code y = a * sin(t)**3 \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Subsection Ellipse \end_layout \begin_layout Standard The ellipse was first studied by Menaechmus \begin_inset LatexCommand cite key "ellipse" \end_inset . Euclid wrote about the ellipse and it was given its present name by Apollonius. The focus and directrix of an ellipse were considered by Pappus. Kepler, in 1602, said he believed that the orbit of Mars was oval, then he later discovered that it was an ellipse with the sun at one focus. In fact Kepler introduced the word \emph on \color black focus \emph default \color inherit and published his discovery in 1609. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/astro.png lyxscale 30 width 4cm \end_inset \begin_inset Graphics filename pics/astropar.png lyxscale 40 width 4cm \end_inset \begin_inset Graphics filename pics/lissa.png lyxscale 40 width 4cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard (a)Output of astro.py (b) astropar.py (c) lissa.py \begin_inset LatexCommand label name "fig:(a)Astro.py" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The Cartesian equation is \begin_inset Formula \begin{equation} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\label{eq:Ellipse-cart}\end{equation} \end_inset \end_layout \begin_layout Standard The parametric equations are \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} x=a\cos(t),y=b\sin(t)\label{eq:Ellipse-Par}\end{equation} \end_inset \end_layout \begin_layout Standard The program \emph on \color black ellipse.py \emph default \color inherit uses the parametric equation to plot the curve. Modifying the parametric equations will result in Lissajous figures. The output of lissa.py are shown in figure \begin_inset LatexCommand ref reference "fig:(a)Astro.py" \end_inset (c). \end_layout \begin_layout Standard \align left \emph on Example ellipse.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 2 \end_layout \begin_layout LyX-Code b = 3 \end_layout \begin_layout LyX-Code t = linspace(0, 2 * pi, 100) \end_layout \begin_layout LyX-Code x = a * sin(t) \end_layout \begin_layout LyX-Code y = b * cos(t) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard \align left \emph on Example lissa.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 2 \end_layout \begin_layout LyX-Code b = 3 \end_layout \begin_layout LyX-Code t= linspace(0, 2*pi,100) \end_layout \begin_layout LyX-Code x = a * sin(2*t) \end_layout \begin_layout LyX-Code y = b * cos(t) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code x = a * sin(3*t) \end_layout \begin_layout LyX-Code y = b * cos(2*t) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard The Lissajous curves are closed if the ratio of the arguments for sine and cosine functions is an integer. Otherwise open curves will result, both are shown in figure \begin_inset LatexCommand ref reference "fig:(a)Astro.py" \end_inset (c). \end_layout \begin_layout Subsection Spirals of Archimedes and Fermat \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/archi.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/fermat.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard (a)Archimedes Spiral (b)Fermat's Spiral (c)Polar Rose \begin_inset LatexCommand label name "fig:(a)Archimedes-Fermat" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The spiral of Archimedes is represented by the equation \begin_inset Formula $r=a\theta$ \end_inset . Fermat's Spiral is given by \begin_inset Formula $r^{2}=a^{2}\theta$ \end_inset . The output of archi.py and fermat.py are shown in figure \begin_inset LatexCommand ref reference "fig:(a)Archimedes-Fermat" \end_inset . \end_layout \begin_layout Standard \align left \emph on Example archi.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 2 \end_layout \begin_layout LyX-Code th= linspace(0, 10*pi,200) \end_layout \begin_layout LyX-Code r = a*th \end_layout \begin_layout LyX-Code polar(th,r) \end_layout \begin_layout LyX-Code axis([0, 2*pi, 0, 70]) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard \align left \emph on Example fermat.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = 2 \end_layout \begin_layout LyX-Code th= linspace(0, 10*pi,200) \end_layout \begin_layout LyX-Code r = sqrt(a**2 * th) \end_layout \begin_layout LyX-Code polar(th,r) \end_layout \begin_layout LyX-Code polar(th, -r) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Subsection Polar Rose \end_layout \begin_layout Standard A rose or rhodonea curve is a sinusoid \begin_inset Formula $r=\cos(k\theta)$ \end_inset plotted in polar coordinates. If k is an even integer, the curve will have \begin_inset Formula $2k$ \end_inset petals and \begin_inset Formula $k$ \end_inset petals if it is odd. If k is rational, then the curve is closed and has finite length. If k is irrational, then it is not closed and has infinite length. \end_layout \begin_layout Standard \align left \emph on Example rose.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code k = 4 \end_layout \begin_layout LyX-Code th = linspace(0, 10*pi,1000) \end_layout \begin_layout LyX-Code r = cos(k*th) \end_layout \begin_layout LyX-Code polar(th,r) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard There are dozens of other famous curves whose details are available on the Internet. It may be an interesting exercise for the reader. For more details refer to \begin_inset LatexCommand cite key "gap-system,wikipedia,wolfram" \end_inset on the Internet. \end_layout \begin_layout Section Power Series \begin_inset LatexCommand label name "sec:Power-Series" \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/series_sin.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/rose.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Outputs of (a)series_sin.py (b) rose.py \begin_inset LatexCommand label name "fig:Functions-Series" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard Trigonometric functions like sine and cosine sounds very familiar to all of us, due to our interaction with them since high school days. However most of us would find it difficult to obtain the numerical values of , say \begin_inset Formula $\sin5^{0}$ \end_inset , without trigonometric tables or a calculator. We know that differentiating a sine function twice will give you the original function, with a sign reversal, which implies \end_layout \begin_layout Standard \begin_inset Formula \[ \frac{d^{2}y}{dx^{2}}+y=0\] \end_inset \end_layout \begin_layout Standard which has a series solution of the form \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} y=a_{0}\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n}}{(2n)!}+a_{1}\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n+1}}{(2n+1)!}\label{eq:Trig Series}\end{equation} \end_inset \end_layout \begin_layout Standard These are the Maclaurin series for sine and cosine functions. The following code plots several terms of the sine series and their sum. \end_layout \begin_layout Standard \align left \emph on Example series_sin.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from scipy import factorial \end_layout \begin_layout LyX-Code x = linspace(-pi, pi, 50) \end_layout \begin_layout LyX-Code y = zeros(50) \end_layout \begin_layout LyX-Code for n in range(5): \end_layout \begin_layout LyX-Code term = (-1)**(n) * (x**(2*n+1)) / factorial(2*n+1) \end_layout \begin_layout LyX-Code y = y + term \end_layout \begin_layout LyX-Code #plot(x,term) #uncomment to see each term \end_layout \begin_layout LyX-Code plot(x, y, '+b') \end_layout \begin_layout LyX-Code plot(x, sin(x),'r') # compare with the real one \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard The output of \emph on \color black series_sin.py \emph default \color inherit is shown in figure \begin_inset LatexCommand ref reference "fig:Functions-Series" \end_inset (a). For comparison the \begin_inset Formula $\sin$ \end_inset function from the library is plotted. The values calculated by using the series becomes closer to the actual value with more and more number of terms. The error can be obtained by adding the following lines to \emph on series_sin.py \emph default and the effect of number of terms on the error can be studied. \end_layout \begin_layout LyX-Code err = y - sin(x) \end_layout \begin_layout LyX-Code plot(x,err) \end_layout \begin_layout LyX-Code for k in err: \end_layout \begin_layout LyX-Code print k \end_layout \begin_layout Section Fourier Series \end_layout \begin_layout Standard A Fourier series is an expansion of a periodic function \begin_inset Formula $f(x)$ \end_inset in terms of an infinite sum of sines and cosines. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. \end_layout \begin_layout Standard The examples below shows how to generate a square wave and sawtooth wave using this technique. To make the output better, increase the number of terms by changing the argument of the range() function, used in the for loop. The output of the programs are shown in figure \begin_inset LatexCommand ref reference "fig:Square-and-Sawtooth" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/sawtooth.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/square.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Sawtooth and Square waveforms generated using Fourier series \begin_inset LatexCommand label name "fig:Square-and-Sawtooth" \end_inset . \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example fourier_square.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code N = 100 # number of points \end_layout \begin_layout LyX-Code x = linspace(0.0, 2 * pi, N) \end_layout \begin_layout LyX-Code y = zeros(N) \end_layout \begin_layout LyX-Code for n in range(5): \end_layout \begin_layout LyX-Code term = sin((2*n+1)*x) / (2*n+1) \end_layout \begin_layout LyX-Code y = y + term \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard \align left \emph on Example fourier_sawtooth.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code N = 100 # number of points \end_layout \begin_layout LyX-Code x = linspace(-pi, pi, N) \end_layout \begin_layout LyX-Code y = zeros(N) \end_layout \begin_layout LyX-Code for n in range(1,10): \end_layout \begin_layout LyX-Code term = (-1)**(n+1) * sin(n*x) / n \end_layout \begin_layout LyX-Code y = y + term \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section 2D plot using colors \end_layout \begin_layout Standard A two dimensional matrix can be represented graphically by assigning a color to each point proportional to the value of that element. The program imshow1.py makes a \begin_inset Formula $50\times50$ \end_inset matrix filled with random numbers and uses \shape italic imshow() \shape default to plot it. The result is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-imshow1.py" \end_inset (a). \end_layout \begin_layout Standard \align left \emph on Example imshow1.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code m = random([50,50]) \end_layout \begin_layout LyX-Code imshow(m) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/imshow.png lyxscale 70 width 4cm \end_inset \begin_inset Graphics filename pics/julia.png lyxscale 40 width 4cm \end_inset \begin_inset Graphics filename pics/mgrid2.png lyxscale 70 width 4cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Outputs of (a) imshow1.py (b) julia.py (c) mgrid2.py \begin_inset LatexCommand label name "fig:Output-of-imshow1.py" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Section Fractals \end_layout \begin_layout Standard Fractals \begin_inset Foot status collapsed \begin_layout Standard http://en.wikipedia.org/wiki/Fractal \end_layout \end_inset are a part of fractal geometry, which is a branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole (e.g.: twigs and tree branches). A fractal is a design of infinite details. It is created using a mathematical formula. No matter how closely you look at a fractal, it never loses its detail. It is infinitely detailed, yet it can be contained in a finite space. Fractals are generally self-similar and independent of scale. The theory of fractals was developed from Benoit Mandelbrot's study of complexity and chaos. Complex numbers are required to compute the Mandelbrot and Julia Set fractals and it is assumed that the reader is familiar with the basics of complex numbers. \end_layout \begin_layout Standard To compute the basic Mandelbrot (or Julia) set one uses the equation \begin_inset Formula $f(z)\rightarrow z^{2}+c$ \end_inset , where both z and c are complex numbers. The function is evaluated in an iterative manner, ie. the result is assigned to \begin_inset Formula $z$ \end_inset and the process is repeated. The purpose of the iteration is to determine the behavior of the values that are put into the function. If the value of the function goes to infinity (practically to some fixed value, like 1 or 2) after few iterations for a particular value of \begin_inset Formula $z$ \end_inset , that point is considered to be outside the Set. A Julia set can be defined as the set of all the complex numbers \begin_inset Formula $(z)$ \end_inset such that the iteration of \begin_inset Formula $f(z)\rightarrow z^{2}+c$ \end_inset is bounded for a particular value of c. \end_layout \begin_layout Standard To generate the fractal the number of iterations required to diverge is calculated for a set of points in the selected region in the complex plane. The number of iterations taken for diverging decides the color of each point. The points that did not diverge, belonging to the set, are plotted with the same color. The program \emph on julia.py \emph default generates a fractal using a julia set. The program creates a 2D array (200 x 200 elements). For our calculations, this array represents a rectangular region on the complex plane centered at the origin whose lower left corner is (-1,-j) and the upper right corner is (1+j). For 200x200 equidistant points in this plane the number of iterations are calculated and that value is given to the corresponding element of the 2D matrix. The plotting is taken care by the imshow function. The output is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-imshow1.py" \end_inset (b). Change the value of \begin_inset Formula $c$ \end_inset and run the program to generate more patterns. The equation also may be changed. \end_layout \begin_layout Standard \align left \emph on Example julia.py \end_layout \begin_layout LyX-Code ''' \end_layout \begin_layout LyX-Code Region of a complex plane ranging from -1 to +1 in both real \end_layout \begin_layout LyX-Code and imaginary axes is represented using a 2D matrix \end_layout \begin_layout LyX-Code having X x Y elements.For X and Y equal to 200,the stepsize \end_layout \begin_layout LyX-Code in the complex plane is 2.0/200 = 0.01. \end_layout \begin_layout LyX-Code The nature of the pattern depends much on the value of c. \end_layout \begin_layout LyX-Code ''' \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code X = 200 \end_layout \begin_layout LyX-Code Y = 200 \end_layout \begin_layout LyX-Code MAXIT = 100 \end_layout \begin_layout LyX-Code MAXABS = 2.0 \end_layout \begin_layout LyX-Code c = 0.02 - 0.8j # The constant in equation z**2 + c \end_layout \begin_layout LyX-Code m = zeros([X,Y]) # A two dimensional array \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def numit(x,y): # number of iterations to diverge \end_layout \begin_layout LyX-Code z = complex(x,y) \end_layout \begin_layout LyX-Code for k in range(MAXIT): \end_layout \begin_layout LyX-Code if abs(z) <= MAXABS: \end_layout \begin_layout LyX-Code z = z**2 + c \end_layout \begin_layout LyX-Code else: \end_layout \begin_layout LyX-Code return k # diverged after k trials \end_layout \begin_layout LyX-Code return MAXIT # did not diverge, \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code for x in range(X): \end_layout \begin_layout LyX-Code for y in range(Y): \end_layout \begin_layout LyX-Code re = 0.01 * x - 1.0 # complex number for \end_layout \begin_layout LyX-Code im = 0.01 * y - 1.0 # this (x,y) coordinate \end_layout \begin_layout LyX-Code m[x][y] = numit(re,im) # get the color for (x,y) \end_layout \begin_layout LyX-Code imshow(m) # Colored plot using the 2D matrix \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section Meshgrids \end_layout \begin_layout Standard in order to make contour and 3D plots, we need to understand the meshgrid. Consider a rectangular area on the X-Y plane. Assume there are m divisions in the X direction and n divisions in the Y direction. We now have a \begin_inset Formula $m\times n$ \end_inset mesh. A meshgrid is the coordinates of a grid in a 2D plane, x coordinates of each mesh point is held in one matrix and y coordinates are held in another. \end_layout \begin_layout Standard The NumPy function meshgrid() creates two 2x2 matrices from two 1D arrays, as shown in the example below. This can be used for plotting surfaces and contours, by assigning a Z coordinat e to every mesh point. \end_layout \begin_layout Standard \align left \emph on Example mgrid1.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code x = arange(0, 3, 1) \end_layout \begin_layout LyX-Code y = arange(0, 3, 1) \end_layout \begin_layout LyX-Code gx, gy = meshgrid(x, y) \end_layout \begin_layout LyX-Code print gx \end_layout \begin_layout LyX-Code print gy \end_layout \begin_layout Standard \align block The outputs are as shown below, gx(i,j) contains the x-coordinate and gx(i,j) contains the y-coordinate of the point (i,j). \end_layout \begin_layout Standard [[0 1 2] \end_layout \begin_layout Standard [0 1 2] \end_layout \begin_layout Standard [0 1 2]] \end_layout \begin_layout Standard [[0 0 0] \end_layout \begin_layout Standard [1 1 1] \end_layout \begin_layout Standard [2 2 2]] \end_layout \begin_layout Standard \align block We can evaluate a function at all points of the meshgrid by passing the meshgrid as an argument. The program mgrid2.py plots the sum of sines of the x and y coordinates, using imshow to get a result as shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-imshow1.py" \end_inset (c). \end_layout \begin_layout Standard \align left \emph on Example mgrid2.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code x = arange(-3*pi, 3*pi, 0.1) \end_layout \begin_layout LyX-Code y = arange(-3*pi, 3*pi, 0.1) \end_layout \begin_layout LyX-Code xx, yy = meshgrid(x, y) \end_layout \begin_layout LyX-Code z = sin(xx) + sin(yy) \end_layout \begin_layout LyX-Code imshow(z) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section 3D Plots \end_layout \begin_layout Standard Matplotlib supports several types of 3D plots, using the Axes3D class. The following three lines of code are required in every program making 3D plots using matplotlib. \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from mpl_toolkits.mplot3d import Axes3D \end_layout \begin_layout LyX-Code ax = Axes3D(figure()) \end_layout \begin_layout Subsection Surface Plots \end_layout \begin_layout Standard The example mgrid2.py is re-written to make a surface plot using the same equation in surface3d.py and the result is shown in figure \begin_inset LatexCommand ref reference "fig:Suface3d and Line3d" \end_inset (a). \end_layout \begin_layout Standard \align left \emph on Example sufrace3d.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from mpl_toolkits.mplot3d import Axes3D \end_layout \begin_layout LyX-Code ax = Axes3D(figure()) \end_layout \begin_layout LyX-Code x = arange(-3*pi, 3*pi, 0.1) \end_layout \begin_layout LyX-Code y = arange(-3*pi, 3*pi, 0.1) \end_layout \begin_layout LyX-Code xx, yy = meshgrid(x, y) \end_layout \begin_layout LyX-Code z = sin(xx) + sin(yy) \end_layout \begin_layout LyX-Code ax.plot_surface(xx, yy, z, cmap=cm.jet, cstride=1) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/surface3d.png width 5cm \end_inset \begin_inset Graphics filename pics/line3d.png width 5cm \end_inset \end_layout \begin_layout Standard Output of (a)surface3d.py (b)line3d.py \begin_inset LatexCommand label name "fig:Suface3d and Line3d" \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Subsection Line Plots \end_layout \begin_layout Standard Example of a line plot is shown in line3d.py along with the output in figure \begin_inset LatexCommand ref reference "fig:Suface3d and Line3d" \end_inset (b). \end_layout \begin_layout Standard \align left \emph on Example line3d.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from mpl_toolkits.mplot3d import Axes3D \end_layout \begin_layout LyX-Code ax = Axes3D(figure()) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code phi = linspace(0, 2*pi, 400) \end_layout \begin_layout LyX-Code x = cos(phi) \end_layout \begin_layout LyX-Code y = sin(phi) \end_layout \begin_layout LyX-Code z = 0 \end_layout \begin_layout LyX-Code ax.plot(x, y, z, label = 'x')# circle \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code z = sin(4*phi) # modulated in z plane \end_layout \begin_layout LyX-Code ax.plot(x, y, z, label = 'x') \end_layout \begin_layout LyX-Code ax.set_xlabel('X') \end_layout \begin_layout LyX-Code ax.set_ylabel('Y') \end_layout \begin_layout LyX-Code ax.set_zlabel('Z') \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard Modify the code to make x = sin(2*phi) to observe Lissajous figures \end_layout \begin_layout Subsection Wire-frame Plots \end_layout \begin_layout Standard Data for a sphere is generated using the outer product of matrices and plotted, by sphere.py. \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code from mpl_toolkits.mplot3d import Axes3D \end_layout \begin_layout LyX-Code ax = Axes3D(figure()) \end_layout \begin_layout LyX-Code phi = linspace(0, 2 * pi, 100) \end_layout \begin_layout LyX-Code theta = linspace(0, pi, 100) \end_layout \begin_layout LyX-Code x = 10 * outer(cos(phi), sin(theta)) \end_layout \begin_layout LyX-Code y = 10 * outer(sin(phi), sin(theta)) \end_layout \begin_layout LyX-Code z = 10 * outer(ones(size(phi)), cos(theta)) \end_layout \begin_layout LyX-Code ax.plot_wireframe(x,y,z, rstride=2, cstride=2) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section Mayavi, 3D visualization \end_layout \begin_layout Standard For more efficient and advanced 3D visualization, use Mayavi that is available on most of the GNU/Linux platforms. Program ylm20.py plots the spherical harmonics \begin_inset Formula $Y_{m}^{l}$ \end_inset for \begin_inset Formula $l=2,m=0$ \end_inset , using mayavi. The plot of \begin_inset Formula $Y_{2}^{0}=\frac{1}{4}\sqrt{\frac{5}{\pi}}(3\cos^{2}\phi-1)$ \end_inset is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-ylm20.py" \end_inset . \end_layout \begin_layout Standard \align left \emph on Example ylm20.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code from enthought.mayavi import mlab \end_layout \begin_layout LyX-Code polar = linspace(0,pi,100) \end_layout \begin_layout LyX-Code azimuth = linspace(0, 2*pi,100) \end_layout \begin_layout LyX-Code phi,th = meshgrid(polar, azimuth) \end_layout \begin_layout LyX-Code r = 0.25 * sqrt(5.0/pi) * (3*cos(phi)**2 - 1) \end_layout \begin_layout LyX-Code x = r*sin(phi)*cos(th) \end_layout \begin_layout LyX-Code y = r*cos(phi) \end_layout \begin_layout LyX-Code z = r*sin(phi)*sin(th) \end_layout \begin_layout LyX-Code mlab.mesh(x, y, z) \end_layout \begin_layout LyX-Code mlab.show() \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/ylm20.png width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of ylm20.py \begin_inset LatexCommand label name "fig:Output-of-ylm20.py" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Section Exercises \end_layout \begin_layout Enumerate Plot a sine wave using markers +, o and x using three different colors. \end_layout \begin_layout Enumerate Plot \begin_inset Formula $\tan\theta$ \end_inset from \begin_inset Formula $\theta$ \end_inset from \begin_inset Formula $-2\pi$ \end_inset to \begin_inset Formula $2\pi$ \end_inset , watch for singular points. \end_layout \begin_layout Enumerate Plot a circle using the polar() function. \end_layout \begin_layout Enumerate Plot the following from the list of Famous curves at reference \begin_inset LatexCommand cite key "gap-system" \end_inset \newline a) \begin_inset Formula $r^{2}=a^{2}\cos2\theta$ \end_inset , Lemniscate of Bernoulli \newline b) \begin_inset Formula $y=\sqrt{2\pi}e^{-x^{2}/2}$ \end_inset Frequency curve \newline c) \begin_inset Formula $a\cosh(x/a)$ \end_inset catenary \newline d) \begin_inset Formula $\sin(a\theta)$ \end_inset for a = 2, 3, and 4. Rhodonea curves \end_layout \begin_layout Enumerate Generate a triangular wave using Fourier series. \end_layout \begin_layout Enumerate Evaluate \begin_inset Formula $y=\sum_{n=1}^{n=\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}$ \end_inset for 10 terms. \end_layout \begin_layout Enumerate Write a Python program to calculate sine function using series expansion and plot it. \end_layout \begin_layout Enumerate Write a Python program to plot \begin_inset Formula $y=5x^{2}+3x+2$ \end_inset (for x from 0 to 5, 20 points),using pylab, with axes and title. Use red colored circles to mark the points. \end_layout \begin_layout Enumerate Write a Python program to plot a Square wave using Fourier series, number of terms should be a variable. \end_layout \begin_layout Enumerate Write a Python program to read the x and y coordinates from a file, in a two column format, and plot them. \end_layout \begin_layout Enumerate Plot \begin_inset Formula $x^{2}+y^{2}+z^{2}=25$ \end_inset using mayavi. \end_layout \begin_layout Enumerate Make a plot z = sin(x) + sin(y) using imshow() , from \begin_inset Formula $-4\pi to4\pi$ \end_inset for both x and y. \end_layout \begin_layout Enumerate Write Python code to plot \begin_inset Formula $y=x^{2}$ \end_inset , with the axes labelled \end_layout \begin_layout Chapter Type setting using LaTeX \end_layout \begin_layout Standard LaTeX is a powerful typesetting system, used for producing scientific and mathematical documents of high typographic quality. LaTeX is not a word processor! Instead, LaTeX encourages authors not to worry too much about the appearance of their documents but to concentrate on getting the right content. You prepare your document using a plain text editor, and the formatting is specified by commands embedded in your document. The appearance of your document is decided by LaTeX, but you need to specify it using some commands. In this chapter, we will discuss some of these commands mainly to typeset mathematical equations. \begin_inset Foot status collapsed \begin_layout Standard http://www.latex-project.org/ \end_layout \begin_layout Standard http://mirror.ctan.org/info/lshort/english/lshort.pdf \end_layout \begin_layout Standard http://en.wikibooks.org/wiki/LaTeX \end_layout \end_inset \end_layout \begin_layout Section Document classes \end_layout \begin_layout Standard LaTeX provides several predefined document classes (book, article, letter, report, etc.) with extensive sectioning and cross-referencing capabilities. Title, chapter, section, subsection, paragraph, subparagraph etc. are specified by commands and it is the job of LaTeX to format them properly. It does the numbering of sections automatically and can generate a table of contents if requested. Figures and tables are also numbered and placed without the user worrying about it. \end_layout \begin_layout Standard The latex source document (the .tex file) is compiled by the latex program to generate a device independent (the .dvi file) output. From that you can generate postscript or PDF versions of the document. We will start with a simple example \shape italic hello.tex \shape default to demonstrate this process. In a line, anything after a % sign is taken as a comment. \end_layout \begin_layout Standard \align left \emph on Example hello.tex \end_layout \begin_layout LyX-Code \backslash documentclass{article} \end_layout \begin_layout LyX-Code \backslash begin{document} \end_layout \begin_layout LyX-Code Small is beautiful. % I am just a comment \end_layout \begin_layout LyX-Code \backslash end{document} \end_layout \begin_layout Standard Compile, view and make a PDF file using the following commands: \end_layout \begin_layout Standard $ latex hello.tex \end_layout \begin_layout Standard $ xdvi hello.dvi \end_layout \begin_layout Standard $ dvipdf hello.dvi \end_layout \begin_layout Standard \align left The output will look like : Small is beautiful. \end_layout \begin_layout Section Modifying Text \end_layout \begin_layout Standard In the next example \shape italic texts.tex \shape default we will demonstrate different types of text. We will \backslash newline or \backslash \backslash to generate a line break. A blank line will start a new paragraph. \end_layout \begin_layout Standard \align left \emph on Example texts.tex \end_layout \begin_layout Quotation \backslash documentclass{article} \end_layout \begin_layout Quotation \backslash begin{document} \end_layout \begin_layout Quotation This is normal text. \end_layout \begin_layout Quotation \backslash newline \end_layout \begin_layout Quotation \backslash textbf{This is bold face text.} \end_layout \begin_layout Quotation \backslash textit{This is italic text.} \backslash \backslash \end_layout \begin_layout Quotation \backslash tiny{This is tiny text.} \end_layout \begin_layout Quotation \backslash large{This is large text.} \end_layout \begin_layout Quotation \backslash underline{This is underlined text.} \end_layout \begin_layout Quotation \backslash end{document} \end_layout \begin_layout Standard \align left Compiling \shape italic texts.tex, \shape default as explained in the previous example, will genearte the following output. \end_layout \begin_layout Standard \lyxline \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash textnormal{This is normal text.} \end_layout \begin_layout Standard \backslash newline \end_layout \begin_layout Standard \backslash textbf{This is bold face text.} \end_layout \begin_layout Standard \backslash textit{This is italic text.} \backslash \backslash \end_layout \begin_layout Standard \backslash tiny{This is tiny text.} \end_layout \begin_layout Standard \backslash large{This is large text.} \end_layout \begin_layout Standard \backslash underline{This is underlined text.} \end_layout \end_inset \end_layout \begin_layout Standard \lyxline \end_layout \begin_layout Section Dividing the document \end_layout \begin_layout Standard A document is generally organized in to sections, subsections, paragraphs etc. and Latex allows us to do this by inserting commands like section subsection etc. If the document class is book, you can have chapters also. There is a command to generate the table of contents from the sectioning information. \begin_inset Foot status collapsed \begin_layout Standard To generate the table of contents, you may have to compile the document two times. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/sections.png width 12cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of sections.tex \begin_inset LatexCommand label name "fig:Output-of-sections.tex" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example sections.tex \end_layout \begin_layout Quotation \backslash documentclass{article} \end_layout \begin_layout Quotation \backslash begin{document} \end_layout \begin_layout Quotation \backslash tableofcontents \end_layout \begin_layout Quotation \backslash section{Animals} \end_layout \begin_layout Quotation This document defines sections. \end_layout \begin_layout Quotation \backslash subsection{Domestic} \end_layout \begin_layout Quotation This document also defines subsections. \end_layout \begin_layout Quotation \backslash subsubsection{cats and dogs} \end_layout \begin_layout Quotation Cats and dogs are Domestic animals. \end_layout \begin_layout Quotation \backslash end{document} \end_layout \begin_layout Standard The output of sections.tex is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-sections.tex" \end_inset . \end_layout \begin_layout Section Environments \end_layout \begin_layout Standard Environments decide the way in which your text is formatted : numbered lists, tables, equations, quotations, justifications, figure, etc. are some of the environments. Environments are defined like : \end_layout \begin_layout Standard \backslash begin{environment_name} your text \backslash end{environment_name} \end_layout \begin_layout Standard \align block The example program \shape italic environ.tex \shape default demonstrates some of the environments. \end_layout \begin_layout Standard \align left \emph on Example environs.tex \end_layout \begin_layout Standard \backslash documentclass{article} \end_layout \begin_layout Standard \backslash begin{document} \end_layout \begin_layout Standard \backslash begin{flushleft} A bulleted list. \backslash end{flushleft} \end_layout \begin_layout Standard \backslash begin{itemize} \backslash item dog \backslash item cat \backslash end{itemize} \end_layout \begin_layout Standard \backslash begin{center} A numbered List. \backslash end{center} \end_layout \begin_layout Standard \backslash begin{enumerate} \backslash item dog \backslash item cat \backslash end{enumerate} \end_layout \begin_layout Standard \backslash begin{flushright} This text is right justified. \backslash end{flushright} \end_layout \begin_layout Standard \backslash begin{quote} \end_layout \begin_layout Standard Any text inside quote \backslash \backslash environment will appe- \backslash \backslash ar as typed. \backslash \backslash \end_layout \begin_layout Standard \backslash end{quote} \end_layout \begin_layout Standard \backslash begin{verbatim} \end_layout \begin_layout Standard x = 1 \end_layout \begin_layout Standard while x <= 10: \end_layout \begin_layout Standard \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ print x * 5 \end_layout \begin_layout Standard \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ x = x + 1 \end_layout \begin_layout Standard \backslash end{verbatim} \end_layout \begin_layout Standard \backslash end{document} \end_layout \begin_layout Standard \align block The enumerate and itemize are used for making numbered and non-numbered lists. Flushleft, flushright and center are used for specifying text justification. Quote and verbatim are used for portions where we do not want LaTeX to do the formatting. The output of environs.tex is shown below. \end_layout \begin_layout Standard \lyxline \end_layout \begin_layout Standard \align block \begin_inset ERT status collapsed \begin_layout Standard \backslash begin{flushleft} A bulleted list. \backslash end{flushleft} \end_layout \begin_layout Standard \backslash begin{itemize} \backslash item dog \backslash item cat \backslash end{itemize} \end_layout \begin_layout Standard \backslash begin{center} A numbered List. \backslash end{center} \end_layout \begin_layout Standard \backslash begin{enumerate} \backslash item dog \backslash item cat \backslash end{enumerate} \end_layout \begin_layout Standard \backslash begin{flushright} This text is right justified. \backslash end{flushright} \end_layout \begin_layout Standard \backslash begin{quote} \end_layout \begin_layout Standard Any text inside quote \backslash \backslash environment will appe- \backslash \backslash ar as typed. \backslash \backslash \end_layout \begin_layout Standard \backslash end{quote} \end_layout \begin_layout Standard \backslash begin{verbatim} \end_layout \begin_layout Standard x = 1 # a Python program \end_layout \begin_layout Standard while x <= 10: \end_layout \begin_layout Standard print x * 5 \end_layout \begin_layout Standard x = x + 1 \end_layout \begin_layout Standard \backslash end{verbatim} \end_layout \end_inset \end_layout \begin_layout Standard \lyxline \end_layout \begin_layout Section Typesetting Equations \end_layout \begin_layout Standard There two ways to typeset mathematical formulae: in-line within a paragraph, or in a separate line. In-line equations are entered between \shape italic two $ symbols \shape default . The equations in a separate line can be done within the \shape italic equation \shape default environment. Both are demonstrated in math1.tex. \shape italic We use the amsmath package in this example. \end_layout \begin_layout Standard \align left \emph on Example math1.tex \end_layout \begin_layout Quotation \backslash documentclass{article} \end_layout \begin_layout Quotation \backslash usepackage{amsmath} \end_layout \begin_layout Quotation \backslash begin{document} \end_layout \begin_layout Quotation The equation $a^2 + b^2 = c^2$ is typeset as inline. \end_layout \begin_layout Quotation The same can be done in a separate line using \end_layout \begin_layout Quotation \backslash begin{equation} \end_layout \begin_layout Quotation a^2 + b^2 = c^2 \end_layout \begin_layout Quotation \backslash end{equation} \end_layout \begin_layout Quotation \backslash end{document} \end_layout \begin_layout Standard The output of this file is shown below. \end_layout \begin_layout Standard \lyxline \begin_inset ERT status collapsed \begin_layout Standard The equation $a^2 + b^2 = c^2$ is typeset as inline. \end_layout \begin_layout Standard The same can be done in a separate line using \end_layout \begin_layout Standard \backslash begin{equation} \end_layout \begin_layout Standard a^2 + b^2 = c^2 \end_layout \begin_layout Standard \backslash end{equation} \end_layout \end_inset \lyxline \end_layout \begin_layout Standard The equation number becomes 5.1 because this happens to be the first equation in chapter 5. \end_layout \begin_layout Subsection Building blocks for typesetting equations \end_layout \begin_layout Standard To typeset equations, we need to know the commands to make constructs like fraction, sqareroot, integral etc. The following list shows several commands and corresponding outputs. For each item, the output of the command, between the two $ signs, is shown on the right side. The reader is expected to insert then inside the body of a document, compile the file and view the output for practicing. \end_layout \begin_layout Enumerate Extra space \begin_inset Foot status collapsed \begin_layout Standard \backslash quad is for inserting space, the size of a \backslash quad corresponds to the width of the character ‘M’ of the current font. Use \backslash qquad for larger space. \end_layout \end_inset : $A \backslash quad B \backslash qquad C$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $A \backslash quad B \backslash qquad C$ \end_layout \end_inset \end_layout \begin_layout Enumerate Greek letters : $ \backslash alpha \backslash beta \backslash gamma \backslash pi$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash alpha \backslash beta \backslash gamma \backslash pi$ \end_layout \end_inset \end_layout \begin_layout Enumerate Subscript and Exponents : $A_n \backslash quad A^m $\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $A_n \backslash quad A^m$ \end_layout \end_inset \end_layout \begin_layout Enumerate Multiple Exponents : $a^b \backslash quad a^{b^c}$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $a^b \backslash quad a^{b^c}$ \end_layout \end_inset \end_layout \begin_layout Enumerate Fractions : $ \backslash frac{3}{5}$ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash frac{3}{5}$ \end_layout \end_inset \end_layout \begin_layout Enumerate Dots : $n! = 1 \backslash cdot 2 \backslash cdots (n-1) \backslash cdot n$ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $n! = 1 \backslash cdot 2 \backslash cdots (n-1) \backslash cdot n$ \end_layout \end_inset \end_layout \begin_layout Enumerate Under/over lines : $0. \backslash overline{3} = \backslash underline{1/3}}$ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $0. \backslash overline{3} = \backslash underline{1/3}$ \end_layout \end_inset \end_layout \begin_layout Enumerate Vectors : $ \backslash vec{a}$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash vec{a}$ \end_layout \end_inset \end_layout \begin_layout Enumerate Functions : $ \backslash sin x + \backslash arctan y$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash sin x + \backslash arctan y$ \end_layout \end_inset \end_layout \begin_layout Enumerate Square root : $ \backslash sqrt{x^2+y^2}$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash sqrt{x^2+y^2}$ \end_layout \end_inset \end_layout \begin_layout Enumerate Higher roots : $z= \backslash sqrt[3]{x^{2} + \backslash sqrt{y}}$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $z= \backslash sqrt[3]{x^{2} + \backslash sqrt{y}}$ \end_layout \end_inset \end_layout \begin_layout Enumerate Equalities : $A \backslash neq B \backslash quad A \backslash approx C$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $A \backslash neq B \backslash quad A \backslash approx C \backslash quad $ \end_layout \end_inset \end_layout \begin_layout Enumerate Arrows : $ \backslash Leftrightarrow \backslash quad \backslash Downarrow$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash Leftrightarrow \backslash quad \backslash Downarrow$ \end_layout \end_inset \end_layout \begin_layout Enumerate Partial derivative : $ \backslash frac{ \backslash partial ^2A}{ \backslash partial x^2}$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash frac{ \backslash partial ^2A}{ \backslash partial x^2}$ \end_layout \end_inset \end_layout \begin_layout Enumerate Summation : $ \backslash sum_{i=1}^n$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash sum_{i=1}^n$ \end_layout \end_inset \end_layout \begin_layout Enumerate Integration : $ \backslash int_0^{ \backslash frac{ \backslash pi}{2} \backslash sin x}$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash int_0^{ \backslash frac{ \backslash pi}{2}} sin x$ \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Enumerate Product : $ \backslash prod_ \backslash epsilon$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash prod_ \backslash epsilon$ \end_layout \end_inset \end_layout \begin_layout Enumerate Big brackets : $ \backslash Big((x+1)(x-1) \backslash Big)^{2}$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash Big((x+1)(x-1) \backslash Big)^{2}$ \end_layout \end_inset \end_layout \begin_layout Enumerate Integral : $ \backslash int_a^b f(x) dx$\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash int _a^b f(x)dx$ \end_layout \end_inset \end_layout \begin_layout Enumerate Operators : $ \backslash pm \backslash div \backslash times \backslash cup \backslash ast \backslash $\InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \InsetSpace ~ \begin_inset ERT status collapsed \begin_layout Standard $ \backslash pm \backslash div \backslash times \backslash cup \backslash ast $ \end_layout \end_inset \end_layout \begin_layout Section Arrays and matrices \end_layout \begin_layout Standard To typeset arrays, use the array environment, that is similar to the tabular environment. Within an array environment, & character separates columns, \backslash \backslash starts a new line. The command \backslash hline inserts a horizontal line. Alignment of the columns is shown inside braces using characters (lcr) and the | symbol is used for adding vertical lines. An example of making a table is shown below. \end_layout \begin_layout Quotation $ \backslash begin{array}{|l|cr|} \backslash hline \end_layout \begin_layout Quotation person & sex & age \backslash \backslash \end_layout \begin_layout Quotation John & male & 20 \backslash \backslash \end_layout \begin_layout Quotation Mary & female & 10 \backslash \backslash \end_layout \begin_layout Quotation Gopal & male & 30 \backslash \backslash \end_layout \begin_layout Quotation \backslash hline \end_layout \begin_layout Quotation \backslash end{array} $ \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard $ \end_layout \begin_layout Standard \backslash begin{array}{|l|cr|} \backslash hline \end_layout \begin_layout Standard person & sex & age \backslash \backslash \end_layout \begin_layout Standard John & male & 7 \backslash \backslash \end_layout \begin_layout Standard Mary & female & 20 \backslash \backslash \end_layout \begin_layout Standard Gopal & male & 30 \backslash \backslash \end_layout \begin_layout Standard \backslash hline \end_layout \begin_layout Standard \backslash end{array} \end_layout \begin_layout Standard $ \end_layout \end_inset \end_layout \begin_layout Standard The first column is left justified, second is centered and the third is right justified (decided by the {|l|cr|}). If you insert a | character between c and r, it will add a vertical line between second and third columns. \end_layout \begin_layout Standard Let us make a matrix using the same command. \end_layout \begin_layout Quotation $ A = \backslash left( \end_layout \begin_layout Quotation \backslash begin{array}{ccc} \end_layout \begin_layout Quotation x_1 & x_2 & \backslash ldots \backslash \backslash \end_layout \begin_layout Quotation y_1 & y_2 & \backslash ldots \backslash \backslash \end_layout \begin_layout Quotation \backslash vdots & \backslash vdots & \backslash ddots \backslash \backslash \end_layout \begin_layout Quotation \backslash end{array} \end_layout \begin_layout Quotation \backslash right) $ \end_layout \begin_layout Standard The output is shown below. The \backslash left( and \backslash right) provides the enclosure. All the columns are centered. We have also used horizontal, vertical and diagonal dots in this example. \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard $ \end_layout \begin_layout Standard A = \backslash left( \end_layout \begin_layout Standard \backslash begin{array}{ccc} \end_layout \begin_layout Standard x_1 & x_2 & \backslash ldots \backslash \backslash \end_layout \begin_layout Standard y_1 & y_2 & \backslash ldots \backslash \backslash \end_layout \begin_layout Standard \backslash vdots & \backslash vdots & \backslash ddots \backslash \backslash \end_layout \begin_layout Standard \backslash end{array} \end_layout \begin_layout Standard \backslash right) $ \end_layout \end_inset \end_layout \begin_layout Section Floating bodies, Inserting Images \end_layout \begin_layout Standard Figures and tables need special treatment, because they cannot be broken across pages. One method would be to start a new page every time a figure or a table is too large to fit on the present page. This approach would leave pages partially empty, which looks very bad. The easiest solution is to \shape italic float \shape default them and let LaTeX decide the position. ( You can influence the placement of the floats using the arguments [htbp], here, top, bottom or special page). Any material enclosed in a figure or table environment will be treated as floating matter. The \shape italic graphicsx \shape default packages is required in this case. \end_layout \begin_layout Standard \backslash usepackage{graphicx} \end_layout \begin_layout Standard \backslash text{Learn how to insert pictures with caption inside the figure environment.} \end_layout \begin_layout Standard \backslash begin{figure}[h] \end_layout \begin_layout Standard \backslash centering \end_layout \begin_layout Standard \backslash includegraphics[width=0.2 \backslash textwidth]{pics/arcs.png} \end_layout \begin_layout Standard \backslash includegraphics[width=0.2 \backslash textwidth]{pics/sawtooth.png} \end_layout \begin_layout Standard \backslash caption{Picture of Arc and Sawtooth, inserted with [h] option.} \end_layout \begin_layout Standard \backslash end{figure} \end_layout \begin_layout Standard \align left The result is shown below. \lyxline \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Standard \backslash textit{Learn how to insert pictures with caption inside the figure environment.} \end_layout \begin_layout Standard \backslash begin{figure}[h] \end_layout \begin_layout Standard \backslash centering \end_layout \begin_layout Standard \backslash includegraphics[width=0.2 \backslash textwidth]{pics/arcs.png} \end_layout \begin_layout Standard \backslash includegraphics[width=0.2 \backslash textwidth]{pics/sawtooth.png} \end_layout \begin_layout Standard \backslash caption{Picture of Arc and Sawtooth, inserted with [h] option.} \end_layout \begin_layout Standard \backslash end{figure} \end_layout \end_inset \lyxline \end_layout \begin_layout Section Example Application \end_layout \begin_layout Standard Latex source code for a simple question paper listed below. \end_layout \begin_layout Standard \align left \emph on Example qpaper.tex \end_layout \begin_layout Quotation \backslash documentclass{article} \end_layout \begin_layout Quotation \backslash usepackage{amsmath} \end_layout \begin_layout Quotation begin{document} \end_layout \begin_layout Quotation \backslash begin{center} \end_layout \begin_layout Quotation \backslash large{ \backslash textbf{Sample Question Paper \backslash \backslash for \backslash \backslash \end_layout \begin_layout Quotation Mathematics using Python}} \end_layout \begin_layout Quotation \backslash end{center} \end_layout \begin_layout Quotation \backslash begin{tabular}{p{8cm}r} \end_layout \begin_layout Quotation \backslash textbf{Duration:3 Hrs} & \backslash textbf{30 weightage} \end_layout \begin_layout Quotation \backslash end{tabular} \backslash \backslash \end_layout \begin_layout Quotation \backslash section{Answer all Questions. $4 \backslash times 1 \backslash frac{1}{2}$} \end_layout \begin_layout Quotation \backslash begin{enumerate} \end_layout \begin_layout Quotation \backslash item What are the main document classes in LaTeX. \end_layout \begin_layout Quotation \backslash item Typeset $ \backslash sin^{2}x+ \backslash cos^{2}x=1$ using LaTeX. \end_layout \begin_layout Quotation \backslash item Plot a circle using the polar() function. \end_layout \begin_layout Quotation \backslash item Write code to print all perfect cubes upto 2000. \end_layout \begin_layout Quotation \backslash end{enumerate} \end_layout \begin_layout Quotation \backslash section{Answer any two Questions. $3 \backslash times 5$} \end_layout \begin_layout Quotation \backslash begin{enumerate} \end_layout \begin_layout Quotation \backslash item Set a sample question paper using LaTeX. \end_layout \begin_layout Quotation \backslash item Using Python calculate the GCD of two numbers \end_layout \begin_layout Quotation \backslash item Write a program with a Canvas and a circle. \end_layout \begin_layout Quotation \backslash end{enumerate} \end_layout \begin_layout Quotation \backslash begin{center} \backslash text{End} \backslash end{center} \end_layout \begin_layout Quotation \backslash end{document} \end_layout \begin_layout Standard The formatted output is shown below. \end_layout \begin_layout Standard \align center \begin_inset Graphics filename pics/qpaper.png width 12cm \end_inset \end_layout \begin_layout Section Exercises \end_layout \begin_layout Enumerate What are the main document classes supported by LaTeX. \end_layout \begin_layout Enumerate How does Latex differ from other word processor programs. \end_layout \begin_layout Enumerate Write a .tex file to typeset 'All types of Text Available' in tiny, large, underline and italic. \end_layout \begin_layout Enumerate Rewrite the previous example to make the output a list of numbered lines. \end_layout \begin_layout Enumerate Generate an article with section and subsections with table of contents. \end_layout \begin_layout Enumerate Typeset 'All types of justifications' to print it three times; left, right and centered. \end_layout \begin_layout Enumerate Write a .tex file that prints 12345 in five lines (one character per line). \end_layout \begin_layout Enumerate Typeset a Python program to generate the multiplication table of 5, using verbatim. \end_layout \begin_layout Enumerate Typeset \begin_inset Formula $\sin^{2}x+\cos^{2}x=1$ \end_inset \end_layout \begin_layout Enumerate Typeset \begin_inset Formula $\left(\sqrt{x^{2}+y^{2}}\right)^{2}=x^{2}+y^{2}$ \end_inset \end_layout \begin_layout Enumerate Typeset \begin_inset Formula $\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{n}$ \end_inset \end_layout \begin_layout Enumerate Typeset \begin_inset Formula $\frac{\partial A}{\partial x}=A$ \end_inset \end_layout \begin_layout Enumerate Typeset \begin_inset Formula $\int_{0}^{\pi}\cos x.dx$ \end_inset \end_layout \begin_layout Enumerate Typeset \begin_inset Formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ \end_inset \end_layout \begin_layout Enumerate Typeset \begin_inset Formula $A=\left(\begin{array}{cc} 1 & 2\\ 3 & 4\end{array}\right)$ \end_inset \end_layout \begin_layout Enumerate Typeset \begin_inset Formula $R=\left(\begin{array}{cc} \sin\theta & \cos\theta\\ \cos\theta & \sin\theta\end{array}\right)$ \end_inset \end_layout \begin_layout Chapter Numerical methods \end_layout \begin_layout Standard Solving mathematical equations is an important requirement for various branches of science but many of them evade an analytic solution. The field of numerical analysis explores the techniques that give approximate but accurate solutions to such problems. \begin_inset Foot status collapsed \begin_layout Standard Introductory methods of numerical analysis by S.S.Sastry \end_layout \begin_layout Standard http://ads.harvard.edu/books/1990fnmd.book/ \end_layout \end_inset Even when they have a solution, for all practical purposes we need to evaluate the numeric value of the result, with the desired accuracy. We will focus on developing simple working programs rather than going into the theoretical details. The mathematical equations giving numerical solutions will be explored by changing various parameters and nature of input data. \end_layout \begin_layout Section Derivative of a function \end_layout \begin_layout Standard The mathematical definition of the derivative of a function \begin_inset Formula $f(x)$ \end_inset at point \begin_inset Formula $x$ \end_inset can be approximated by equation \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} {lim\atop \Delta x\rightarrow0}\frac{f(x+\frac{\triangle x}{2})-f(x-\frac{\triangle x}{2})}{\triangle x}\label{eq:derivative}\end{equation} \end_inset \end_layout \begin_layout Standard neglecting the higher order terms. The accuracy of the derivative calculated using discrete values \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none depends on the stepsize \begin_inset Formula $\triangle x$ \end_inset . It will also depends on the number of higher order derivatives the function has. We will try to explore these aspects using the program \family default \series default \shape default \size default \emph on \bar default \noun default \color black diff.py \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none , which evaluates the derivatives of few functions using two different stepsizes (0.1 ans 0.01). The input values to function deriv() are the function to be differentiated, the point at which the derivative is to be found and the stepsize \begin_inset Formula $\triangle x$ \end_inset . \end_layout \begin_layout Standard \align left \emph on Example diff.py \end_layout \begin_layout LyX-Code def f1(x): \end_layout \begin_layout LyX-Code return x**2 \end_layout \begin_layout LyX-Code def f2(x): \end_layout \begin_layout LyX-Code return x**4 \end_layout \begin_layout LyX-Code def f3(x): \end_layout \begin_layout LyX-Code return x**10 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def deriv(func, x, dx=0.1): \end_layout \begin_layout LyX-Code df = func(x+dx/2)-func(x-dx/2) \end_layout \begin_layout LyX-Code return df/dx \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print deriv(f1, 1.0), deriv(f1, 1.0, 0.01) \end_layout \begin_layout LyX-Code print deriv(f2, 1.0), deriv(f2, 1.0, 0.01) \end_layout \begin_layout LyX-Code print deriv(f3, 1.0), deriv(f3, 1.0, 0.01) \end_layout \begin_layout Standard The output of the program is shown below. Comparing the two numbers on the same line shows the effect of stepsize. Comparing the first number on each row shows the effect of the number of higher order derivatives the function has. For the same stepsize \begin_inset Formula $x^{4}$ \end_inset gives much less error than \begin_inset Formula $x^{10}$ \end_inset . Second derivative of \begin_inset Formula $x^{2}$ \end_inset is constant and the result becomes exact, result on the first line. \end_layout \begin_layout Quotation 2.0 2.0 \end_layout \begin_layout Quotation 4.01 4.0001 \end_layout \begin_layout Quotation 10.3015768754 10.0030001575 \end_layout \begin_layout Standard You may explore other functions by modifying the program. \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none It can be seen that the function deriv(), evaluates the function at two points to calculate the derivative. The higher order terms can be calculated by evaluating the function at more points. Techniques used for this will be discussed in section \begin_inset LatexCommand ref reference "sec:Interpolation" \end_inset , on interpolation. \end_layout \begin_layout Subsection Differentiate Sine to get Cosine \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/vdiff.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/vdiff_bigerror.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Outputs of vdiff.py (a)for \begin_inset Formula $\triangle x=0.005$ \end_inset (b) for \begin_inset Formula $\triangle x=1.0$ \end_inset \begin_inset LatexCommand label name "fig:Outputs-of-vdiff.py" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The program \emph on \color black diff.py \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none in the previous example can only calculate the value of the derivative at a given point. In the program \family default \series default \shape default \size default \emph on \bar default \noun default \color black vdiff.py \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none , we use a vectorized version of our deriv() function. The defined function is sine and the derivative is calculated using the vectorized version of deriv(). The actual cosine function also is plotted for comparison. The output of vdiff.py is shown in \begin_inset LatexCommand ref reference "fig:Outputs-of-vdiff.py" \end_inset (a). \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none The value of \begin_inset Formula $\triangle x$ \end_inset is increased to 1.0 \family default \series default \shape default \size default \emph default \bar default \noun default \color inherit by changing one line of code as \begin_inset Formula $y=vecderiv(x,1.0)$ \end_inset and the result is shown in \begin_inset LatexCommand ref reference "fig:Outputs-of-vdiff.py" \end_inset (b). The values calculated using our function is shown using \begin_inset Formula $+$ \end_inset marker, while the continuous curve is the expected result , ie. the cosine curve. \end_layout \begin_layout Standard \align left \emph on Example vdiff.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code def f(x): \end_layout \begin_layout LyX-Code return sin(x) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def deriv(x,dx=0.005): \end_layout \begin_layout LyX-Code df = f(x+dx/2)-f(x-dx/2) \end_layout \begin_layout LyX-Code return df/dx \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code vecderiv = vectorize(deriv) \end_layout \begin_layout LyX-Code x = linspace(-2*pi, 2*pi, 200) \end_layout \begin_layout LyX-Code y = vecderiv(x) \end_layout \begin_layout LyX-Code plot(x,y,'+') \end_layout \begin_layout LyX-Code plot(x,cos(x)) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section Numerical Integration \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Standard \align center \begin_inset Graphics filename pics/integ1.png lyxscale 50 width 5cm \end_inset \begin_inset Graphics filename pics/integ2.png lyxscale 50 width 5cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Area under the curve is divided it in to a large number of intervals. Area of each of them is calculated by assuming them to be trapezoids. \begin_inset LatexCommand label name "fig:Integration" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral. The objective is to find the area under the curve as shown in figure \begin_inset LatexCommand ref reference "fig:Integration" \end_inset . One method is to divide this area in to large number of sub-intervals and find the sum of their areas. The interval \begin_inset Formula $a\leq x\leq b$ \end_inset is divided in to \begin_inset Formula $n$ \end_inset sub-intervals, each of length \begin_inset Formula $h=(b-a)/n$ \end_inset , and area of a sub-interval is approximated by \begin_inset Formula \[ \int_{x_{n-1}}^{x_{n}}ydx=\frac{h}{2}(y_{n-1}+y_{n})\] \end_inset the integral is given by \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \int_{a}^{b}ydx=\frac{h}{2}\left[y_{0}+2(y_{1}+y_{2}+\ldots+y_{n-1})+y_{n}\right]\label{eq:Trapez}\end{equation} \end_inset This is the sum of the areas of the individual trapezoids. The error in using the trapezoid rule is approximately proportional to \begin_inset Formula $1/n^{2}$ \end_inset . If the number of sub-intervals is doubled, the error is reduced by a factor of 4. The program \emph on trapez.py \emph default does integration of a given function using equation \begin_inset LatexCommand ref reference "eq:Trapez" \end_inset . We will choose an example where the results can be cross checked easily, the value of \begin_inset Formula $\pi$ \end_inset is calculated by evaluating the area of a unit circle by integrating the equation of a circle. \end_layout \begin_layout Standard \align left \emph on Example trapez.py \end_layout \begin_layout LyX-Code from math import * \end_layout \begin_layout LyX-Code def y(x): # equation of a circle \end_layout \begin_layout LyX-Code return sqrt(1.0 - x**2) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def trapez(f, a, b, n): \end_layout \begin_layout LyX-Code h = (b-a) / n \end_layout \begin_layout LyX-Code sum = 0 \end_layout \begin_layout LyX-Code x = 0.5 * h # f(x) at middle of the slice \end_layout \begin_layout LyX-Code for i in range (1,n): \end_layout \begin_layout LyX-Code sum = sum + h * f(x) \end_layout \begin_layout LyX-Code x = x + h \end_layout \begin_layout LyX-Code return sum \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print 4 * trapez(y, 0.0, 1.0,1000) \end_layout \begin_layout LyX-Code print 4 * trapez(y, 0.0, 1.0,10000) \end_layout \begin_layout LyX-Code print trapez(sin,0,2,1000) # Why the error ? \end_layout \begin_layout Standard The output is shown below. The result gets better by increasing \begin_inset Formula $n$ \end_inset thus resulting in smaller \begin_inset Formula $h$ \end_inset . The last line shows, how things can go wrong if the arguments are given in the integer format. Learn how to avoid such pitfalls while writing programs. It is left as an exercise to the reader to modify the function trapez() to accept integer arguments also. \end_layout \begin_layout Standard 3.14041703178 \end_layout \begin_layout Standard 3.14155546691 \end_layout \begin_layout Standard 0.0 \end_layout \begin_layout Section Ordinary Differential Equations \end_layout \begin_layout Standard Differential equations are one of the most important mathematical tools used in producing models for physical and biological processes. In this section, we will discuss the numerical methods for solving the initial value problem for first-order ordinary differential equations. Consider the equation, \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{dy}{dx}=f(x,y);\,\,\,\, y(x_{0})=y_{0}\label{eq:ODE}\end{equation} \end_inset \end_layout \begin_layout Standard where the derivative of the function \begin_inset Formula $f(x,y)$ \end_inset is known and the value of the function at some value of \begin_inset Formula $x=x_{0}$ \end_inset also is known. The objective is to find out the value of the function for other values of \begin_inset Formula $x$ \end_inset . The underlying idea of any routine for solving the initial value problem is to rewrite the \begin_inset Formula $dy$ \end_inset and \begin_inset Formula $dx$ \end_inset as finite steps \begin_inset Formula $\triangle y$ \end_inset and \begin_inset Formula $\triangle x$ \end_inset , and multiply the equations by \begin_inset Formula $\triangle x$ \end_inset . This gives algebraic formulas for the change in the value of \begin_inset Formula $y(x)$ \end_inset when \begin_inset Formula $x$ \end_inset is changed by one stepsize \begin_inset Formula $\triangle x$ \end_inset . In the limit of making the stepsize very small, a good approximation to the underlying differential equation is achieved. \end_layout \begin_layout Standard Implementation of this procedure results in the Euler’s method, which is conceptually very important, but not recommended for any practical use. In this section we will discuss Euler's method and the Runge-Kutta method with the help of example programs. For detailed information refer to \begin_inset LatexCommand cite key "numerical recepies,mathcs.emory" \end_inset . \end_layout \begin_layout Subsection Euler method \end_layout \begin_layout Standard The equations of Euler's method can be obtained as follows. By the definition of derivative, \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} y^{'}(x_{n},y_{n})={lim\atop h\rightarrow0}\frac{y(x_{n}+h)-y(x_{n})}{h}\label{eq:Euler}\end{equation} \end_inset \end_layout \begin_layout Standard For sufficiently small values of \begin_inset Formula $h$ \end_inset , we can write, \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} y(x_{n}+h)=y(x_{n},y_{n})+hy^{'}(x_{n})\label{eq:Euler2}\end{equation} \end_inset \end_layout \begin_layout Standard The above equations implies that, if the value of the function \begin_inset Formula $y(x)$ \end_inset is known to be \begin_inset Formula $y_{n}$ \end_inset at the point \begin_inset Formula $x_{n}$ \end_inset , its value at a nearby point \begin_inset Formula $x_{n+1}$ \end_inset is given by \begin_inset Formula $y_{n}+h\times y^{'}.$ \end_inset The program \emph on euler.py \emph default calculates the value of sine function using its derivative, ie. the cosine function. We start from \begin_inset Formula $x=0$ \end_inset , where \begin_inset Formula $\sin(x)=0$ \end_inset and compute the subsequent values using the derivative, \begin_inset Formula $cos(x)$ \end_inset , and compare the result with the actual sine function. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/euler.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/rkmethod.png width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard (a)Output of euler.py (b)Four intermediate steps of RK4 method \begin_inset LatexCommand label name "fig:Output-of-euler.py" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example euler.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code h = 0.01 # stepsize \end_layout \begin_layout LyX-Code x = 0.0 # initial values \end_layout \begin_layout LyX-Code y = 0.0 \end_layout \begin_layout LyX-Code ax = [] # Lists to store x and y \end_layout \begin_layout LyX-Code ay = [] \end_layout \begin_layout LyX-Code while x < 2*pi: \end_layout \begin_layout LyX-Code y = y + h * math.cos(x) # Euler equation \end_layout \begin_layout LyX-Code x = x + h \end_layout \begin_layout LyX-Code ax.append(x) \end_layout \begin_layout LyX-Code ay.append(y) \end_layout \begin_layout LyX-Code plot(ax,ay) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard The output of \emph on euler.py \emph default is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-euler.py" \end_inset . \end_layout \begin_layout Subsection Runge-Kutta method \end_layout \begin_layout Standard The formula \begin_inset LatexCommand ref reference "eq:Euler" \end_inset used by Euler method which advances a solution from \begin_inset Formula $x_{n}tox_{n+1}$ \end_inset is not symmetric, it advances the solution through an interval \begin_inset Formula $h$ \end_inset , but uses derivative information only at the beginning of that interval. Better results are obtained if we take \shape italic trial \shape default step to the midpoint of the interval and use the value of both \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset at that midpoint to compute the \shape italic real \shape default step across the whole interval. This is called the second-order Runge-Kutta or the midpoint method. This procedure can be further extended to higher orders. \end_layout \begin_layout Standard The fourth order Runge-Kutta method is the most popular one and is commonly referred as the Runge-Kutta method. In each step the derivative is evaluated four times as shown in figure \begin_inset LatexCommand ref reference "fig:Outputs-of-(a)rk4.py" \end_inset (a). Once at the initial point, twice at trial midpoints, and once at a trial endpoint. Every trial evaluation uses the value of the function from the previous trial point, ie. \begin_inset Formula $k_{2}$ \end_inset is evaluated using \begin_inset Formula $k_{1}$ \end_inset and not using \begin_inset Formula $y_{n}$ \end_inset . From these derivatives the final function value is calculated, The calculation is done using the equations, \end_layout \begin_layout Standard \align center \begin_inset Formula $\begin{array}{c} k_{1}=hf(x_{n},y_{n})\\ k_{2}=hf(x_{n}+\frac{h}{2},y_{n}+\frac{k_{1}}{2})\\ k_{3}=hf(x_{n}+\frac{h}{2},y_{n}+\frac{k_{2}}{2})\\ k_{4}=hf(x_{n}+h,y_{n}+k_{3})\end{array}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} y_{n+1}=y_{n}+\frac{1}{6}\left(k_{1}+2k_{2}+2k_{3}+k_{4}\right)\label{eq:Runge-Kutta4}\end{equation} \end_inset \end_layout \begin_layout Standard The program rk4.py listed below uses the equations shown above to calculate the sine function, by integrating the cosine. The output is shown in figure \begin_inset LatexCommand ref reference "fig:Outputs-of-(a)rk4.py" \end_inset (a). \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/rk4.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/eurk4.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Outputs of (a)rk4.py (b) compareEuRK4.py \begin_inset LatexCommand label name "fig:Outputs-of-(a)rk4.py" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example rk4.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def rk4(x, y, fx, h = 0.1): # x, y , f(x), stepsize \end_layout \begin_layout LyX-Code k1 = h * fx(x) \end_layout \begin_layout LyX-Code k2 = h * fx(x + h/2.0) \end_layout \begin_layout LyX-Code k3 = h * fx(x + h/2.0) \end_layout \begin_layout LyX-Code k4 = h * fx(x + h) \end_layout \begin_layout LyX-Code return y + ( k1/6 + k2/3 + k3/3 + k4/6 ) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code h = 0.01 # stepsize \end_layout \begin_layout LyX-Code x = 0.0 # initial values \end_layout \begin_layout LyX-Code y = 0.0 \end_layout \begin_layout LyX-Code ax = [x] \end_layout \begin_layout LyX-Code ay = [y] \end_layout \begin_layout LyX-Code while x < math.pi: \end_layout \begin_layout LyX-Code y = rk4(x,y,math.cos) \end_layout \begin_layout LyX-Code x = x + h \end_layout \begin_layout LyX-Code ax.append(x) \end_layout \begin_layout LyX-Code ay.append(y) \end_layout \begin_layout LyX-Code plot(ax,ay) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard The program compareEuRK4.py calculates the values of Sine by integrating Cosine. The errors in both cases are evaluated at every step, by comparing it with the sine function, and plotted as shown in figure \begin_inset LatexCommand ref reference "fig:Outputs-of-(a)rk4.py" \end_inset (b). The accuracy of Runge-Kutta method is far superior to that of Euler's method, for the same step size. \end_layout \begin_layout Standard \align left \emph on Example compareEuRK4.py \end_layout \begin_layout LyX-Code from scipy import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def rk4(x, y, fx, h = 0.1): # x, y , f(x), stepsize \end_layout \begin_layout LyX-Code k1 = h * fx(x) \end_layout \begin_layout LyX-Code k2 = h * fx(x + h/2.0) \end_layout \begin_layout LyX-Code k3 = h * fx(x + h/2.0) \end_layout \begin_layout LyX-Code k4 = h * fx(x + h) \end_layout \begin_layout LyX-Code return y + ( k1/6 + k2/3 + k3/3 + k4/6 ) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code h = 0.1 # stepsize \end_layout \begin_layout LyX-Code x = 0.0 # initial values \end_layout \begin_layout LyX-Code ye = 0.0 # for Euler \end_layout \begin_layout LyX-Code yr = 0.0 # for RK4 \end_layout \begin_layout LyX-Code ax = [] # Lists to store results \end_layout \begin_layout LyX-Code euerr = [] \end_layout \begin_layout LyX-Code rkerr = [] \end_layout \begin_layout LyX-Code while x < 2*pi: \end_layout \begin_layout LyX-Code ye = ye + h * math.cos(x) # Euler method \end_layout \begin_layout LyX-Code yr = rk4(x, yr, cos, h) # RK4 method \end_layout \begin_layout LyX-Code x = x + h \end_layout \begin_layout LyX-Code ax.append(x) \end_layout \begin_layout LyX-Code euerr.append(ye - sin(x)) \end_layout \begin_layout LyX-Code rkerr.append(yr - sin(x)) \end_layout \begin_layout LyX-Code plot(ax,euerr,'o') \end_layout \begin_layout LyX-Code plot(ax, rkerr,'+') \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Subsection Function depending on the integral \end_layout \begin_layout Standard In the previous section, the program \shape italic rk4.py \shape default implemented a simplified version of the Runge-Kutta method, the function was assumed to depend on the independent variable only. The program \shape italic rk4_proper.py \shape default listed below implements it properly. The functions \begin_inset Formula $f(x,y)=1+y^{2}$ \end_inset and \begin_inset Formula $f(x,y)=(y-x)/y+x)$ \end_inset are used for testing. Readers may verify the results by manual computing. \end_layout \begin_layout Standard \align left \emph on Example rk4_proper.py \end_layout \begin_layout LyX-Code def f1(x,y): \end_layout \begin_layout LyX-Code return 1 + y**2 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def f2(x,y): \end_layout \begin_layout LyX-Code return (y-x)/(y+x) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def rk4(x, y, fxy, h): # x, y , f(x,y), step \end_layout \begin_layout LyX-Code k1 = h * fxy(x, y) \end_layout \begin_layout LyX-Code k2 = h * fxy(x + h/2.0, y+k1/2) \end_layout \begin_layout LyX-Code k3 = h * fxy(x + h/2.0, y+k2/2) \end_layout \begin_layout LyX-Code k4 = h * fxy(x + h, y+k3) \end_layout \begin_layout LyX-Code ny = y + ( k1/6 + k2/3 + k3/3 + k4/6 ) \end_layout \begin_layout LyX-Code #print x,y,k1,k2,k3,k4, ny \end_layout \begin_layout LyX-Code return ny \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code h = 0.2 # stepsize \end_layout \begin_layout LyX-Code x = 0.0 # initial values \end_layout \begin_layout LyX-Code y = 0.0 \end_layout \begin_layout LyX-Code print rk4(x,y, f1, h) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code h = 1 \end_layout \begin_layout LyX-Code x = 0.0 # initial values \end_layout \begin_layout LyX-Code y = 1.0 \end_layout \begin_layout LyX-Code print rk4(x,y,f2,h) \end_layout \begin_layout Standard The results are shown below. \end_layout \begin_layout Standard 0.202707408081 \end_layout \begin_layout Standard 1.5056022409 \end_layout \begin_layout Section Polynomials \begin_inset LatexCommand label name "sec:Polynomials" \end_inset \end_layout \begin_layout Standard A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable with constant coefficients is given by \begin_inset Formula \begin{equation} a_{n}x^{n}+...+a_{2}x^{2}+a_{1}x+a_{0}\label{eq:Polinomial}\end{equation} \end_inset \end_layout \begin_layout Standard The derivative of \begin_inset LatexCommand ref reference "eq:Polinomial" \end_inset is, \end_layout \begin_layout Standard \begin_inset Formula \[ na_{n}x^{n-1}+...+2a_{2}x+a_{1}\] \end_inset \end_layout \begin_layout Standard It is easy to find the derivative of a polynomial. Complicated functions can be analyzed by approximating them with polynomials. Taylor's theorem states that any sufficiently smooth function can locally be approximated by a polynomial. Computers use this property to evaluate trigonometric, logarithmic and exponential functions. \end_layout \begin_layout Standard One dimensional polynomials can be explored using the \begin_inset Formula $poly1d$ \end_inset function from Numpy. You can define a polynomial by supplying the coefficient as a list. For example , the statement p = poly1d([3,4,7]) constructs the polynomial \begin_inset Formula $3x^{2}+4x+7$ \end_inset . Numpy supports several polynomial operations. The following example demonstrates evaluation at a particular value, multiplica tion, differentiation, integration and division of polynomials using \shape italic poly1d \shape default . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/polyplot.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of polyplot.py \begin_inset LatexCommand label name "fig:Output-of-polyplot.py" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example poly.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code a = poly1d([3,4,5]) \end_layout \begin_layout LyX-Code b = poly1d([6,7]) \end_layout \begin_layout LyX-Code c = a * b + 5 \end_layout \begin_layout LyX-Code d = c/a \end_layout \begin_layout LyX-Code print a \end_layout \begin_layout LyX-Code print a(0.5) \end_layout \begin_layout LyX-Code print b \end_layout \begin_layout LyX-Code print a * b \end_layout \begin_layout LyX-Code print a.deriv() \end_layout \begin_layout LyX-Code print a.integ() \end_layout \begin_layout LyX-Code print d[0], d[1] \end_layout \begin_layout Standard The output of \emph on \color black poly.py \emph default \color inherit is shown below. \end_layout \begin_layout Standard \begin_inset Formula $\begin{array}[b]{l} 3x^{2}+4x+5\\ 7.75\\ 6x+7\\ 18x^{3}+45x^{2}+58x+35\\ 6x+4\\ 1x^{3}+2x^{2}+5x\\ 6x+7\\ 5\end{array}$ \end_inset \end_layout \begin_layout Standard The last two lines show the result of the polynomial division, quotient and reminder. Note that a polynomial can take an array argument for evaluation to return the results in an array. The program \emph on \color black polyplot.py \emph default \color inherit evaluates polynomial \begin_inset LatexCommand ref reference "eq:polynomial of Sine" \end_inset and its first derivative. \begin_inset Formula \begin{equation} x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}\label{eq:polynomial of Sine}\end{equation} \end_inset \end_layout \begin_layout Standard The results are shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-polyplot.py" \end_inset . The equation \begin_inset LatexCommand ref reference "eq:polynomial of Sine" \end_inset is the first four terms of series representing sine wave (7! = 5040). The derivative looks like cosine as expected. Try adding more terms and change the limits to see the effects. \end_layout \begin_layout Standard \align left \emph on Example polyplot.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code x = linspace(-pi, pi, 100) \end_layout \begin_layout LyX-Code a = poly1d([-1.0/5040,0,1.0/120,0,-1.0/6,0,1,0]) \end_layout \begin_layout LyX-Code da = a.deriv() \end_layout \begin_layout LyX-Code y = a(x) \end_layout \begin_layout LyX-Code y1 = da(x) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code plot(x,y1) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Subsection Taylor's Series \begin_inset LatexCommand label name "sub:Taylor's-Series" \end_inset \end_layout \begin_layout Standard If a function and its derivatives are known at some point \begin_inset Formula $x=a$ \end_inset , we can express \begin_inset Formula $f(x)$ \end_inset in the vicinity of that point using a polynomial. The Taylor series expansion is given by, \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} f(x)=f(a)+(x-a)f^{'}(a)+\frac{(x-a)^{2}}{2!}f^{''}(a)+\cdots+\frac{(x-a)^{n}}{n!}f^{n}(a)\label{eq:Taylor's Series}\end{equation} \end_inset \end_layout \begin_layout Standard For example let us consider the equation \begin_inset Formula \begin{equation} f(x)=x^{3}+x^{2}+x\label{eq:xcube}\end{equation} \end_inset \end_layout \begin_layout Standard We can see that \begin_inset Formula $f(0)=0$ \end_inset and the derivatives are \begin_inset Formula \[ f'(x)=3x^{2}+2x+1;\,\,\,\,\,\,\,\,\,\,\, f''(x)=6x+2;\,\,\,\,\,\, f'''(x)=6\] \end_inset \end_layout \begin_layout Standard Using the values of the derivatives at \begin_inset Formula $x=0$ \end_inset and equation \begin_inset LatexCommand ref reference "eq:Taylor's Series" \end_inset , we evaluate the function at \begin_inset Formula $x=.5$ \end_inset , using the polynomial expression, \end_layout \begin_layout Standard \begin_inset Formula \[ f(0.5)=0+0.5\times1+\frac{0.5^{2}\times2}{2!}+\frac{0.5^{3}\times6}{3!}=.875\] \end_inset \end_layout \begin_layout Standard The result is same as \begin_inset Formula $0.5^{3}+0.5^{2}+0.5=.875$ \end_inset . We have calculated it manually for \begin_inset Formula $x=.5$ \end_inset . We can also do this using Numpy as shown in the program taylor.py. \end_layout \begin_layout Standard \align left \emph on Example taylor.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code p = poly1d([1,1,1,0]) \end_layout \begin_layout LyX-Code dp = p.deriv() \end_layout \begin_layout LyX-Code dp2 = dp.deriv() \end_layout \begin_layout LyX-Code dp3 = dp2.deriv() \end_layout \begin_layout LyX-Code a = 0 # The known point \end_layout \begin_layout LyX-Code x = 0 # Evaluate at x \end_layout \begin_layout LyX-Code while x < .5: \end_layout \begin_layout LyX-Code tay = p(a) + (x-a)* dp(a) + \backslash \end_layout \begin_layout LyX-Code (x-a)**2 * dp2(a) / 2 + (x-a)**3 * dp3(a)/6 \end_layout \begin_layout LyX-Code print '%5.1f %8.5f \backslash t%8.5f'%(x, p(x), tay) \end_layout \begin_layout LyX-Code x = x + .1 \end_layout \begin_layout Standard The result is shown below. \end_layout \begin_layout Quotation 0.0 0.00000 0.00000 \end_layout \begin_layout Quotation 0.1 0.11100 0.11100 \end_layout \begin_layout Quotation 0.2 0.24800 0.24800 \end_layout \begin_layout Quotation 0.3 0.41700 0.41700 \end_layout \begin_layout Quotation 0.4 0.62400 0.62400 \end_layout \begin_layout Subsection Sine and Cosine Series \end_layout \begin_layout Standard In the equation \begin_inset LatexCommand ref reference "eq:Taylor's Series" \end_inset , let us choose \begin_inset Formula $f(x)=sin(x)$ \end_inset and \begin_inset Formula $a=0$ \end_inset . If \begin_inset Formula $a=0$ \end_inset , then the series is known as the Maclaurin Series. The first term will become \begin_inset Formula $sin(0)$ \end_inset , which is just zero. The other terms involve the derivatives of \begin_inset Formula $sin(x)$ \end_inset . The first, second and third derivatives of \begin_inset Formula $sin(x)$ \end_inset are \begin_inset Formula $cos(x)$ \end_inset , \begin_inset Formula $-sin(x)$ \end_inset and \begin_inset Formula $-cos(x)$ \end_inset , respectively. Evaluating each of these at zero, we get 1, 0 and -1 respectively. The terms with even powers vanish, resulting in, \begin_inset Formula \begin{equation} \sin(x)=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\cdots\,\,\,\,\,=\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n+1}}{(2n+1)!}\label{eq:Taylor Sine}\end{equation} \end_inset \end_layout \begin_layout Standard We can find the cosine series in a similar manner, to get \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \cos(x)=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\cdots\,\,\,\,\,=\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n}}{(2n)!}\label{eq:Taylor cosine}\end{equation} \end_inset \end_layout \begin_layout Standard The program series_sc.py evaluates the sine and cosine series. The output is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-series_sc.py" \end_inset . Compare the output of polyplot.py from section \begin_inset LatexCommand ref reference "sec:Polynomials" \end_inset with this. In both cases, we have evaluated the polynomial of sine function. In the present case, we can easily modify the number of terms and the logic is simpler. \end_layout \begin_layout Standard \align left \emph on Example series_sc.py \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/series_sc.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of series_sc.py \begin_inset LatexCommand label name "fig:Output-of-series_sc.py" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def f(n): # Factorial function \end_layout \begin_layout LyX-Code if n == 0: \end_layout \begin_layout LyX-Code return 1 \end_layout \begin_layout LyX-Code else: \end_layout \begin_layout LyX-Code return n * f(n-1) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code NP = 100 \end_layout \begin_layout LyX-Code x = linspace(-pi, pi, NP) \end_layout \begin_layout LyX-Code sinx = zeros(NP) \end_layout \begin_layout LyX-Code cosx = zeros(NP) \end_layout \begin_layout LyX-Code for n in range(10): \end_layout \begin_layout LyX-Code sinx += (-1)**(n) * (x**(2*n+1)) / f(2*n+1) \end_layout \begin_layout LyX-Code cosx += (-1)**(n) * (x**(2*n)) / f(2*n) \end_layout \begin_layout LyX-Code plot(x, sinx) \end_layout \begin_layout LyX-Code plot(x, cosx,'r') \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Section Finding roots of an equation \end_layout \begin_layout Standard In general, an equation may have any number of roots, or no roots at all. For example \begin_inset Formula $f(x)=x^{2}$ \end_inset has a single root whereas \begin_inset Formula $f(x)=sin(x)$ \end_inset has an infinite number of roots. The roots can be located visually, by looking at the intersections with the x-axis. Another useful tool for detecting and bracketing roots is the incremental search method. The basic idea behind the incremental search method is simple: if \begin_inset Formula $f(x1)$ \end_inset and \begin_inset Formula $f(x2)$ \end_inset have opposite signs, then there is at least one root in the interval \begin_inset Formula $(x1,x2)$ \end_inset . If the interval is small enough, it is likely to contain a single root. Thus the zeroes of \begin_inset Formula $f(x)$ \end_inset can be detected by evaluating the function at intervals of \begin_inset Formula $\Delta x$ \end_inset and looking for change in sign. \end_layout \begin_layout Standard There are several potential problems with the incremental search method: It is possible to miss two closely spaced roots if the search increment \begin_inset Formula $\Delta x$ \end_inset is larger than the spacing of the roots. Certain singularities of \begin_inset Formula $f(x)$ \end_inset can be mistaken for roots. For example, \begin_inset Formula $f(x)=tan(x)$ \end_inset changes sign at odd multiples of \begin_inset Formula $\pi/2$ \end_inset , but these locations are not true zeroes as shown in figure \begin_inset LatexCommand ref reference "fig:NRplot" \end_inset (b). \end_layout \begin_layout Standard Example \shape italic rootsearch.py \shape default implements the function \begin_inset Formula $root()$ \end_inset that searches the roots of a function \begin_inset Formula $f(x)$ \end_inset from \begin_inset Formula $x=a$ \end_inset to \begin_inset Formula $x=b$ \end_inset , incrementing it by \begin_inset Formula $dx$ \end_inset . \end_layout \begin_layout Standard \align left \emph on Example rootsearch.py \end_layout \begin_layout LyX-Code def func(x): \end_layout \begin_layout LyX-Code return x**3-10.0*x*x + 5 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def root(f,a,b,dx): \end_layout \begin_layout LyX-Code x = a \end_layout \begin_layout LyX-Code while True: \end_layout \begin_layout LyX-Code f1 = f(x) \end_layout \begin_layout LyX-Code f2 = f(x+dx) \end_layout \begin_layout LyX-Code if f1*f2 < 0: \end_layout \begin_layout LyX-Code return x, x + dx \end_layout \begin_layout LyX-Code x = x + dx \end_layout \begin_layout LyX-Code if x >= b: \end_layout \begin_layout LyX-Code return (None,None) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code x,y = root(func, 0.0, 1.0,.1) \end_layout \begin_layout LyX-Code print x,y \end_layout \begin_layout LyX-Code x,y = root(math.cos, 0.0, 4,.1) \end_layout \begin_layout LyX-Code print x,y \end_layout \begin_layout Standard The outputs are (0.7 , 0.8) and (1.5 , 1.6). The root of cosine, \begin_inset Formula $\pi/2$ \end_inset , is between 1.5 and 1.6. After the root has been located roughly, we can find the root with any specified accuracy, using bisection method, Newton-Raphson method etc. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/nrplot.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/tanx.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard (a)Function \begin_inset Formula $2x^{2}-3x-5$ \end_inset and its tangents at \begin_inset Formula $x=4$ \end_inset and \begin_inset Formula $x=4$ \end_inset (b) tan(x) \begin_inset LatexCommand label name "fig:NRplot" \end_inset . \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Method of Bisection \end_layout \begin_layout Standard The method of bisection finds the root by successively halving the interval until it becomes sufficiently small. Bisection is not the fastest method available for computing roots, but it is the most reliable. Once a root has been bracketed, bisection will always find it. The method of bisection works in the following manner. If there is a root between \begin_inset Formula $x1$ \end_inset and \begin_inset Formula $x2$ \end_inset , then \begin_inset Formula $f(x1)\times f(x2)<0$ \end_inset . Next, we compute \begin_inset Formula $f(x3)$ \end_inset , where \begin_inset Formula $x3=(x1+x2)/2$ \end_inset . If \begin_inset Formula $f(x2)\times f(x3)<0$ \end_inset , then the root must be in \begin_inset Formula $(x2,x3)$ \end_inset and we replace the original bound \begin_inset Formula $x1$ \end_inset by \begin_inset Formula $x3$ \end_inset . Otherwise, the root lies between \begin_inset Formula $x1$ \end_inset and \begin_inset Formula $x3$ \end_inset , in that case \begin_inset Formula $x2$ \end_inset is replaced by \begin_inset Formula $x3$ \end_inset . This process is repeated until the interval has been reduced to the specified value, say \begin_inset Formula $\varepsilon$ \end_inset . \end_layout \begin_layout Standard The number of bisections required to reach a prescribed limit, \begin_inset Formula $\varepsilon,$ \end_inset is given by equation \begin_inset LatexCommand ref reference "eq:bisection" \end_inset . \begin_inset Formula \begin{equation} n=\frac{\ln(|\triangle x|)/\varepsilon}{\ln2}\label{eq:bisection}\end{equation} \end_inset \end_layout \begin_layout Standard The program \shape italic bisection.py \shape default finds the root of the equation \begin_inset Formula $x^{3}-10x^{2}+5$ \end_inset . The starting values are found using the program \shape italic rootsearch.py \shape default . The results are printed for two different accuracies. \end_layout \begin_layout Standard \align left \emph on Example bisection.py \end_layout \begin_layout LyX-Code import math def func(x): \end_layout \begin_layout LyX-Code return x**3 - 10.0* x*x + 5 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def bisect(f, x1, x2, epsilon=1.0e-9): \end_layout \begin_layout LyX-Code f1 = f(x1) \end_layout \begin_layout LyX-Code f2 = f(x2) \end_layout \begin_layout LyX-Code if f1*f2 > 0.0: \end_layout \begin_layout LyX-Code print 'x1 and x2 are on the same side of x-axis' \end_layout \begin_layout LyX-Code return \end_layout \begin_layout LyX-Code n = math.ceil(math.log(abs(x2 - x1)/epsilon)/math.log(2.0)) \end_layout \begin_layout LyX-Code n = int(n) \end_layout \begin_layout LyX-Code for i in range(n): \end_layout \begin_layout LyX-Code x3 = 0.5 * (x1 + x2) \end_layout \begin_layout LyX-Code f3 = f(x3) \end_layout \begin_layout LyX-Code if f3 == 0.0: return x3 \end_layout \begin_layout LyX-Code if f2*f3 < 0.0: \end_layout \begin_layout LyX-Code x1 = x3 \end_layout \begin_layout LyX-Code f1 = f3 \end_layout \begin_layout LyX-Code else: \end_layout \begin_layout LyX-Code x2 = x3 \end_layout \begin_layout LyX-Code f2 = f3 \end_layout \begin_layout LyX-Code return (x1 + x2)/2.0 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print bisect(func, 0.70, 0.8, 1.0e-4) \end_layout \begin_layout LyX-Code print bisect(func, 0.70, 0.8, 1.0e-9) \end_layout \begin_layout Subsection Newton-Raphson Method \end_layout \begin_layout Standard The Newton–Raphson algorithm requires the derivative of the function also to evaluate the roots. Therefore, it is usable only in problems where \begin_inset Formula $f'(x)$ \end_inset can be readily computed. It does not require the value at two points to start with. We start with an initial guess which is reasonably close to the true root. Then the function is approximated by its tangent line and the x-intercept of the tangent line is calculated. This value is taken as the next guess and the process is repeated. The Newton-Raphson formula is shown below. \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} x_{i+1}=x_{i}-\frac{f(x_{i})}{f'(x_{i})}\label{eq:Newton-Raphson}\end{equation} \end_inset \end_layout \begin_layout Standard Figure \begin_inset LatexCommand ref reference "fig:NRplot" \end_inset (a) shows the graph of the quadratic equation \begin_inset Formula $2x^{2}-3x-5=0$ \end_inset and its two tangents. It can be seen that the zeros are at x = -1 and x = 2.5, and we use the program newraph.py shown below to find the roots. The function nr() is called twice, and we get the roots nearer to the correspon ding starting values. \end_layout \begin_layout Standard \align left \emph on Example newraph.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code def f(x): \end_layout \begin_layout LyX-Code return 2.0 * x**2 - 3*x - 5 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def df(x): \end_layout \begin_layout LyX-Code return 4.0 * x - 3 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def nr(x, tol = 1.0e-9): \end_layout \begin_layout LyX-Code for i in range(30): \end_layout \begin_layout LyX-Code dx = -f(x)/df(x) \end_layout \begin_layout LyX-Code #print x \end_layout \begin_layout LyX-Code x = x + dx \end_layout \begin_layout LyX-Code if abs(dx) < tol: \end_layout \begin_layout LyX-Code return x \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code print nr(4) \end_layout \begin_layout LyX-Code print nr(0) \end_layout \begin_layout Standard The output is shown below. \end_layout \begin_layout Standard 2.5 \end_layout \begin_layout Standard -1.0 \end_layout \begin_layout Standard \align block Uncomment the print statement inside nr() to view how fast this method converges , compared to the bisection method. The program newraph_plot.py, listed below is used for generating the figure \begin_inset LatexCommand ref reference "fig:NRplot" \end_inset . \end_layout \begin_layout Standard \align left \emph on Example newraph_plot.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code def f(x): \end_layout \begin_layout LyX-Code return 2.0 * x**2 - 3*x - 5 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def df(x): \end_layout \begin_layout LyX-Code return 4.0 * x - 3 \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code vf = vectorize(f) \end_layout \begin_layout LyX-Code x = linspace(-2, 5, 100) \end_layout \begin_layout LyX-Code y = vf(x) \end_layout \begin_layout LyX-Code # Tangents at x=3 and 4, using one point slope formula \end_layout \begin_layout LyX-Code x1 = 4 \end_layout \begin_layout LyX-Code tg1 = df(x1)*(x-x1) + f(x1) \end_layout \begin_layout LyX-Code x1 = 3 \end_layout \begin_layout LyX-Code tg2 = df(x1)*(x-x1) + f(x1) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code grid(True) \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code plot(x,tg1) \end_layout \begin_layout LyX-Code plot(x,tg2) \end_layout \begin_layout LyX-Code ylim([-20,40]) \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard \align left We have defined the function \begin_inset Formula $f(x)=2x^{2}-3x-5$ \end_inset and vectorized it. The derivative \begin_inset Formula $4x^{2}-3$ \end_inset also is defined by \begin_inset Formula $df(x)$ \end_inset , which is the slope of \begin_inset Formula $f(x)$ \end_inset . The tangents are drawn at \begin_inset Formula $x=4$ \end_inset and \begin_inset Formula $x=3$ \end_inset , using the point slope formula for a line \begin_inset Formula $y=m(x-x1)+y1$ \end_inset . \end_layout \begin_layout Section System of Linear Equations \end_layout \begin_layout Standard A system of \begin_inset Formula $m$ \end_inset linear equations with \begin_inset Formula $n$ \end_inset unknowns can be written in a matrix form and can be solved by using several standard techniques like Gaussian elimination. In this section, the matrix inversion method, using Numpy, is demonstrated. For more information see reference \begin_inset LatexCommand cite key "Kiusalas" \end_inset . \end_layout \begin_layout Subsection Equation solving using matrix inversion \end_layout \begin_layout Standard Non-homogeneous matrix equations of the form \begin_inset Formula $Ax=b$ \end_inset can be solved by matrix inversion to obtain \begin_inset Formula $x=A^{-1}b$ \end_inset . The system of equations \end_layout \begin_layout Standard \align center \begin_inset Formula $\begin{array}[b]{r} 4x+y\,-2z=0\\ 2x-3y+3z=9\\ -6x-2y\,+z=0\end{array}$ \end_inset \end_layout \begin_layout Standard can be represented in the matrix form as \end_layout \begin_layout Standard \align center \begin_inset Formula \begin{eqnarray*} \left(\begin{array}{rrr} 4 & 1 & -2\\ 2 & -3 & 3\\ -6 & -2 & 1\end{array}\right)\left(\begin{array}{c} x\\ y\\ z\end{array}\right) & = & \left(\begin{array}{c} 0\\ 9\\ 0\end{array}\right)\end{eqnarray*} \end_inset \end_layout \begin_layout Standard and can be solved by finding the inverse of the coefficient matrix. \end_layout \begin_layout Standard \begin_inset Formula \[ \left(\begin{array}{c} x\\ y\\ z\end{array}\right)=\left(\begin{array}{rrr} 4 & 1 & -2\\ 2 & -3 & 3\\ -6 & -2 & 1\end{array}\right)^{-1}\left(\begin{array}{c} 0\\ 9\\ 0\end{array}\right)\] \end_inset \end_layout \begin_layout Standard Using numpy we can solve this as shown in solve_eqn.py \end_layout \begin_layout Standard \align left Example solve_eqn.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code b = array([0,9,0]) \end_layout \begin_layout LyX-Code A = array([ [4,1,-2], [2,-3,3],[-6,-2,1]]) \end_layout \begin_layout LyX-Code print dot(linalg.inv(A),b) \end_layout \begin_layout Standard The result will be [ 0.75 -2. 0.5 ], that means \begin_inset Formula $x=0.75,y=-2,z=0.5$ \end_inset . This can be verified by substituting them back in to the equations. \end_layout \begin_layout Standard Exercise: solve x+y+3z = 6; 2x+3y-4z=6;3x+2y+7z=0 \end_layout \begin_layout Section Least Squares Fitting \end_layout \begin_layout Standard A mathematical procedure for finding the best-fitting curve \begin_inset Formula $f(x)$ \end_inset for a given set of points \begin_inset Formula $(x_{n},y_{n})$ \end_inset by minimizing the sum of the squares of the vertical offsets of the points from the curve is called least squares fitting. The least square fit is obtained by minimizing the function, \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} S(a_{0},a_{1},\ldots,a_{m})=\sum_{i=0}^{n}\left[y_{i}-f(x_{i})\right]^{2}\label{eq:Least square}\end{equation} \end_inset \end_layout \begin_layout Standard with respect to each \begin_inset Formula $a_{i}$ \end_inset and the condition for that is \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} \frac{\partial S}{\partial a_{i}}=0,\,\,\,\,\, i=0,1,\ldots m\label{eq:LS2}\end{equation} \end_inset \end_layout \begin_layout Standard For a linear fit, the equation is \begin_inset Formula \[ f(a,b)=a+bx\] \end_inset \end_layout \begin_layout Standard Solving the equations \begin_inset Formula $\frac{\partial S}{\partial a}=0$ \end_inset and \begin_inset Formula $\frac{\partial S}{\partial b}=0$ \end_inset will give the result, \begin_inset Formula \begin{equation} b=\frac{\sum y_{i}(x-\overline{x})}{\sum x_{i}(x-\overline{x})},\,\,\,\, and\,\,\,\, a=\overline{y}-\overline{x}b\label{eq:LS3}\end{equation} \end_inset \end_layout \begin_layout Standard where \begin_inset Formula $\overline{x}$ \end_inset and \begin_inset Formula $\overline{y}$ \end_inset are the mean values defined by the equations, \begin_inset Formula \begin{equation} \overline{x}=\frac{1}{n+1}\sum_{i=0}^{n}x_{i},\,\,\,\,\,\,\overline{y}=\frac{1}{n+1}\sum_{i=0}^{n}y{}_{i}\label{eq:LS4}\end{equation} \end_inset \end_layout \begin_layout Standard The program lsfit.py demonstrates the usage of equations \begin_inset LatexCommand ref reference "eq:LS3" \end_inset and \begin_inset LatexCommand ref reference "eq:LS4" \end_inset . \end_layout \begin_layout Standard \align left \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/lsfit.png lyxscale 50 width 7cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of lsfit.py \begin_inset LatexCommand label name "fig:Output-of-lsfit.py" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example lsfit.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code NP = 50 \end_layout \begin_layout LyX-Code r = 2*ranf([NP]) - 0.5 \end_layout \begin_layout LyX-Code x = linspace(0,10,NP) \end_layout \begin_layout LyX-Code data = 3 * x + 2 + r \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code xbar = mean(x) \end_layout \begin_layout LyX-Code ybar = mean(data) \end_layout \begin_layout LyX-Code b = sum(data*(x-xbar)) / sum(x*(x-xbar)) \end_layout \begin_layout LyX-Code a = ybar - xbar * b \end_layout \begin_layout LyX-Code print a,b \end_layout \begin_layout LyX-Code y = a + b * x \end_layout \begin_layout LyX-Code plot(x,y) \end_layout \begin_layout LyX-Code plot(x,data,'ob') \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard The raw data is made by adding random numbers (between -1 and 1) to the \begin_inset Formula $y$ \end_inset coordinates generated by \begin_inset Formula $y=3*x+2$ \end_inset . The Numpy functions mean() and sum() are used. The output is shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-lsfit.py" \end_inset . \end_layout \begin_layout Section Interpolation \begin_inset LatexCommand label name "sec:Interpolation" \end_inset \end_layout \begin_layout Standard Interpolation is the process of constructing a function \begin_inset Formula $f(x)$ \end_inset from a set of data points \begin_inset Formula $(x_{i},y_{i})$ \end_inset , in the interval \begin_inset Formula $a \begin_inset Text \begin_layout Standard \begin_inset Formula $x_{0}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $x_{0}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{0}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $x_{1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $y{}_{1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula $\left[y_{1}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{0},y_{1}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $x_{2}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $y_{2}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{2}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{1},y_{2}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{0},y_{1},y_{2}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $x_{3}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $y_{3}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{3}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{2},y_{3}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{1},y_{2},y_{3}\right]$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\left[y_{0},y_{1},y_{2},y_{3}\right]$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Standard \align center \begin_inset Tabular \begin_inset Text \begin_layout Standard 0 \end_layout \end_inset \begin_inset Text \begin_layout Standard 0 \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard 1 \end_layout \end_inset \begin_inset Text \begin_layout Standard 3 \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\frac{3-0}{1-0}=3$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard 2 \end_layout \end_inset \begin_inset Text \begin_layout Standard 14 \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\frac{14-3}{2-1}=11$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\frac{11-3}{2-0}=4$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \end_layout \end_inset \begin_inset Text \begin_layout Standard 3 \end_layout \end_inset \begin_inset Text \begin_layout Standard 39 \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\frac{39-14}{3-2}=25$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\frac{25-11}{3-1}=7$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula $\frac{7-4}{3-0}=1$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Standard The table given above shows the divided difference table for the data set x = [0,1,2,3] and y = [0,3,14,39], calculated manually. The program newpoly.py can be used for calculating the coefficients, which prints the output [0, 3, 4, 1]. \end_layout \begin_layout Standard \align left \emph on Example newpoly.py \end_layout \begin_layout LyX-Code from copy import copy \end_layout \begin_layout LyX-Code def coef(x,y): \end_layout \begin_layout LyX-Code a = copy(y) \end_layout \begin_layout LyX-Code m = len(x) \end_layout \begin_layout LyX-Code for k in range(1,m): \end_layout \begin_layout LyX-Code tmp = copy(a) \end_layout \begin_layout LyX-Code for i in range(k,m): \end_layout \begin_layout LyX-Code tmp[i] = (a[i] - a[i-1])/(x[i]-x[i-k]) \end_layout \begin_layout LyX-Code a = copy(tmp) \end_layout \begin_layout LyX-Code return a \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code x = [0,1,2,3] \end_layout \begin_layout LyX-Code y = [0,3,14,39] \end_layout \begin_layout LyX-Code print coef(x,y) \end_layout \begin_layout Standard We start by copying the list \begin_inset Formula $y$ \end_inset to coefficient \begin_inset Formula $a$ \end_inset , the first element \begin_inset Formula $a_{0}=y_{0}$ \end_inset . While calculating the differences, we have used two loops and a temporary list. The same can be done in a better way using arrays of Numpy \begin_inset Foot status collapsed \begin_layout Standard This function is from reference \begin_inset LatexCommand cite key "Kiusalas" \end_inset , some PDF versions of this book are available on the web. \end_layout \end_inset , as shown in newpoly2.py. \end_layout \begin_layout Standard \align left \emph on Example newpoly2.py \end_layout \begin_layout LyX-Code from numpy import * \end_layout \begin_layout LyX-Code def coef(x,y): \end_layout \begin_layout LyX-Code a = copy(y) \end_layout \begin_layout LyX-Code m = len(x) \end_layout \begin_layout LyX-Code for k in range(1,m): \end_layout \begin_layout LyX-Code a[k:m] = (a[k:m] - a[k-1])/(x[k:m]-x[k-1]) \end_layout \begin_layout LyX-Code return a \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code x = array([0,1,2,3]) \end_layout \begin_layout LyX-Code y = array([0,3,14,39]) \end_layout \begin_layout LyX-Code print coef(x,y) \end_layout \begin_layout Standard The next step is to calculate the value of \begin_inset Formula $y$ \end_inset for any given value of \begin_inset Formula $x$ \end_inset , using the coefficients already calculated. The program \shape italic newinterpol.py \shape default calculates the coefficients using the four data points. The function eval() uses the recurrence relation \end_layout \begin_layout Standard \begin_inset Formula \begin{equation} P_{k}(x)=a_{n-k}+(x-x_{n-k})P_{k-1}(x),\,\,\,\, k=1,2,\ldots n\label{eq:NPrecur}\end{equation} \end_inset \end_layout \begin_layout Standard The program generates 20 new values of \begin_inset Formula $x$ \end_inset , and calculate corresponding values of \begin_inset Formula $y$ \end_inset and plots them along with the original data points, as shown in figure \begin_inset LatexCommand ref reference "fig:Output-of-newton_in3.py" \end_inset . \end_layout \begin_layout Standard \align left \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/newton_in3.png lyxscale 50 width 8cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Output of newton_in3.py \begin_inset LatexCommand label name "fig:Output-of-newton_in3.py" \end_inset \end_layout \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \align left \emph on Example newinterpol.py \end_layout \begin_layout LyX-Code from pylab import * \end_layout \begin_layout LyX-Code def eval(a,xpoints,x): \end_layout \begin_layout LyX-Code n = len(xpoints) - 1 \end_layout \begin_layout LyX-Code p = a[n] \end_layout \begin_layout LyX-Code for k in range(1,n+1): \end_layout \begin_layout LyX-Code p = a[n-k] + (x -xpoints[n-k]) * p \end_layout \begin_layout LyX-Code return p \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code def coef(x,y): \end_layout \begin_layout LyX-Code a = copy(y) \end_layout \begin_layout LyX-Code m = len(x) \end_layout \begin_layout LyX-Code for k in range(1,m): \end_layout \begin_layout LyX-Code a[k:m] = (a[k:m] - a[k-1])/(x[k:m]-x[k-1]) \end_layout \begin_layout LyX-Code return a \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code x = array([0,1,2,3]) \end_layout \begin_layout LyX-Code y = array([0,3,14,39]) \end_layout \begin_layout LyX-Code coef = coef(x,y) \end_layout \begin_layout LyX-Code \end_layout \begin_layout LyX-Code NP = 20 \end_layout \begin_layout LyX-Code newx = linspace(0,3, NP) # New x-values \end_layout \begin_layout LyX-Code newy = zeros(NP) \end_layout \begin_layout LyX-Code for i in range(NP): # evaluate y-values \end_layout \begin_layout LyX-Code newy[i] = eval(coef, x, newx[i]) \end_layout \begin_layout LyX-Code plot(newx, newy,'-x') \end_layout \begin_layout LyX-Code plot(x, y,'ro') \end_layout \begin_layout LyX-Code show() \end_layout \begin_layout Standard You may explore the results for new points outside the range by changing the second argument of line \shape italic newx = linspace(0,3,NP) \shape default to a higher value. \end_layout \begin_layout Standard Look for similarities between Taylor's series discussed in section \begin_inset LatexCommand ref reference "sub:Taylor's-Series" \end_inset that and polynomial interpolation process. The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. The finite difference is the discrete analog of the derivative. Using the divided difference method, we are in fact calculating the derivatives in the discrete form. \end_layout \begin_layout Section Exercises \end_layout \begin_layout Enumerate Differentiate \begin_inset Formula $5x^{2}+3x+5$ \end_inset numerically and evaluate at \begin_inset Formula $x=2$ \end_inset and \begin_inset Formula $x=-2$ \end_inset . \end_layout \begin_layout Enumerate Write code to numerically differentiate \begin_inset Formula $\sin(x^{2})$ \end_inset and plot it by vectorizing the function. Compare with the analytical result. \end_layout \begin_layout Enumerate Integrate \begin_inset Formula $\ln x$ \end_inset , \begin_inset Formula $e^{x}$ \end_inset from \begin_inset Formula $x=1$ \end_inset to \begin_inset Formula $2$ \end_inset . \end_layout \begin_layout Enumerate Solve \begin_inset Formula $2x+y=3;-x+4y=0;3+3y=-1$ \end_inset using matrices. \end_layout \begin_layout Enumerate Modify the program julia.py, \begin_inset Formula $c=0.2-0.8j$ \end_inset and \begin_inset Formula $z=z^{6}+c$ \end_inset \end_layout \begin_layout Enumerate Write Python code, using pylab, to solve the following equations using matrices \newline \begin_inset Formula $\begin{array}[b]{r} 4x+y\,-2z=0\\ 2x-3y+3z=9\\ -6x-2y\,+z=0\end{array}$ \end_inset \end_layout \begin_layout Enumerate Find the roots of \begin_inset Formula $5x^{2}+3x-6$ \end_inset using bisection method. \end_layout \begin_layout Enumerate Find the all the roots of \begin_inset Formula $sin(x)$ \end_inset between 0 and 10, using Newton-Raphson method. \end_layout \begin_layout Chapter* Appendix A : Installing GNU/Linux \begin_inset ERT status collapsed \begin_layout Standard \backslash addcontentsline{toc}{chapter}{Appendix A} \end_layout \end_inset \begin_inset ERT status collapsed \begin_layout Standard \backslash markboth{Appendix A}{Installing Ubuntu} \end_layout \end_inset \end_layout \begin_layout Standard Programming can be learned better by practicing and it requires an operating system, and Python interpreter along with some library modules. All these requirements are packaged on the Live CD comes along with this book. You can boot any PC from this CD and start working. However, it is better to install the whole thing to a harddisk. The following section explains howto install GNU/Linux. We have selected the Ubuntu distribution due to its relatively simple installat ion procedure, ease of maintenanace and support for most of the hardware available in the market. \end_layout \begin_layout Section Installing Ubuntu \end_layout \begin_layout Standard Most of the users prefer a dual boot system, to keep their MSWindows working. We will explain the installation process keeping that handicap in mind. All we need is an empty partition of minimum 5 GB size to install Ubuntu. Free space inside a Windows partition will not do, we need to format the partition to install Ubuntu. The Ubuntu installer will make the system multi-boot by searching through all the partitions for installed operating systems. \end_layout \begin_layout Standard \align left \shape italic \bar under The System \end_layout \begin_layout Standard This section will describe how Ubuntu was installed on a system, with MSWindows, having the following partitions: \end_layout \begin_layout Standard C: (GNU/Linux calls it /dev/sda1) 20 GB \end_layout \begin_layout Standard D: ( /dev/sda5) 20 GB \end_layout \begin_layout Standard E: (/dev/sda6) 30 GB \end_layout \begin_layout Standard We will use the partition E: to install Ubuntu, it will be formatted. \end_layout \begin_layout Standard \align left \shape italic \bar under The Procedure \end_layout \begin_layout Standard Set the first boot device CD ROM from the BIOS. Boot the PC from the Ubuntu installation CD, we have used Phoenix Live CD (a modified version on Ubuntu 9.1). After 2 to 3 minutes a desktop as shown below will appear. Click on the Installer icon, the window shown next will pop up. Screens will appear to select the language, time zone and keyboard layout as shown in the figures below. \end_layout \begin_layout Standard \align center \begin_inset Graphics filename pics/desktop.png lyxscale 30 width 6cm \end_inset \begin_inset Graphics filename pics/inst1.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard \align center \begin_inset Graphics filename pics/inst2.png lyxscale 50 width 6cm \end_inset \begin_inset Graphics filename pics/inst3.png lyxscale 50 width 6cm \end_inset \end_layout \begin_layout Standard Now we proceed to the important part, choosing a partition to install Ubuntu. \end_layout \begin_layout Standard \align center \begin_inset Graphics filename pics/inst_disk1.png lyxscale 50 width 12cm \end_inset \end_layout \begin_layout Standard The bar on the top graphically displays the existing partitions. Below that there are three options provided : \end_layout \begin_layout Enumerate Install them side by side. \end_layout \begin_layout Enumerate Erase and use the entire disk. \end_layout \begin_layout Enumerate Specify partitions manually. \end_layout \begin_layout Standard If you choose the first option, the Installer will resize and repartition the disk to make some space for the new system. By default this option is marked as selected. The bar at the bottom shows the proposed partition scheme. In the present example, the installer plans to divide the C: drive in to two partitions to put ubuntu on the second. \end_layout \begin_layout Standard We are going to choose the third option, choose the partition manually. We will use the last partition (drive E: ) for installing Ubuntu. Once you choose that and click forward, a screen will appear where we can add, delete and change partitions. We have selected the third partition and clicked on Change. A pop-up window appeared. Using that we selected the file-system type to ext3, marked the format option, and selected the mount point as / . The screen with the pop-up window is shown below. \end_layout \begin_layout Standard \align center \begin_inset Graphics filename pics/inst_disk2.png lyxscale 50 width 12cm \end_inset \end_layout \begin_layout Standard \align left If we proceed with this, a warning will appear complaining about the absence of swap partitions. The swap partition is used for supplementing the RAM with some virtual memory. When RAM is full, processes started but not running will be swapped out. One can install a system without swap partition but it is a good idea to have one. \end_layout \begin_layout Standard We decide to go back on the warning, to delete the E: drive, create two new partitions in that space and make one of them as swap. This also demonstrates how to make new partitions. The screen after deleting E: , with the pop-up window to make the swap partition is shown below. \end_layout \begin_layout Standard \align center \begin_inset Graphics filename pics/inst_makeswap.png lyxscale 50 width 12cm \end_inset \end_layout \begin_layout Standard We made a 2 GB swap. The remaining space is used for making one more partition, as shown in the figure \begin_inset LatexCommand ref reference "fig:Making-the-partition" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/inst_disk3.png lyxscale 50 width 12cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Making the partition to install Ubuntu \begin_inset LatexCommand label name "fig:Making-the-partition" \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard Once disk partitioning is over, you will be presented with a screen to enter a user name and password. \begin_inset Foot status collapsed \begin_layout Standard All GNU/Linux installations ask for a root password during installation. For simplicity, Ubuntu has decided to hide this information. The first user created during installation has special privileges and that password is asked, instead of the root password, for all system administration jobs, like installing new software. \end_layout \end_inset A warning will be issued if the password is less than 8 characters in length. You will be given an option to import desktop settings from other installations already on the disk, choose this if you like. The next screen will confirm the installation. After the installation is over, mat take 10 to 15 minutes, you will be prompted to reboot the system. On rebooting you will be presented with a menu, to choose the operating system to boot. First item in the menu will be the newly installed Ubuntu. \end_layout \begin_layout Section Package Management \end_layout \begin_layout Standard The Ubuntu install CD contains some common application programs like web browser, office package, document viewer, image manipulation program etc. After installing Ubuntu, you may want to add more applications. The Ubuntu repository has an enormous number of packages, that can be installed very easily. You need to have a reasonably fast Internet connection for this purpose. \end_layout \begin_layout Standard From the main menu, open System->Administration->Synaptic package manager. After providing the pass word (of the first user, created during installation), the synaptic window will popup as shown in figure \begin_inset LatexCommand ref reference "fig:Synaptic-package-manager" \end_inset . \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status collapsed \begin_layout Standard \align center \begin_inset Graphics filename pics/synaptic1.png lyxscale 50 width 12cm \end_inset \end_layout \begin_layout Standard \begin_inset Caption \begin_layout Standard Synaptic package manager window \begin_inset LatexCommand label name "fig:Synaptic-package-manager" \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \end_layout \end_inset \end_layout \begin_layout Standard Select Settings->Repositories to get a pop-up window as shown below. Tick the four repositories, close the pop-up window and Click on Reload. Synaptic will now try to download the index files from all these repositories. It may take several minute. \end_layout \begin_layout Standard \align center \begin_inset Graphics filename pics/synaptic2.png lyxscale 50 width 12cm \end_inset \end_layout \begin_layout Standard Now, you are ready to install any package from the Ubuntu repository. Search for any package by name, from these repositories, and install it. If you have done the installation from original Ubuntu CD, you may require the following packges: \end_layout \begin_layout Itemize lyx : A latex front-end. Latex will be installed since lyx dpends on latex. \end_layout \begin_layout Itemize python-matplotlib : The graphics library \end_layout \begin_layout Itemize python-visual : 3D graphics \end_layout \begin_layout Itemize python-imaging-tk : Tkinter, Python Imaging Library etc. will be installed \end_layout \begin_layout Itemize build-essential : C compiler and related tools. \end_layout \begin_layout Subsection Install from repository CD \end_layout \begin_layout Standard We can also install packages from repository CDs. Insert the CD in the drive and Select Add CDROM from the Edit menu of Synaptic. Now all the packages on the CD will be available for search and install. \end_layout \begin_layout Subsubsection Installing from the Terminal \end_layout \begin_layout Standard You can install packages from the terminal also. You have to become the root user by giving the sudo command; \end_layout \begin_layout Standard $ sudo -s \end_layout \begin_layout Standard enter password : \end_layout \begin_layout Standard # \end_layout \begin_layout Standard Note that the prompt changes from $ to #, when you become root. \end_layout \begin_layout Standard Install packages using the command : \end_layout \begin_layout Standard #apt-cdrom add \end_layout \begin_layout Standard #apt-get install mayavi2 \end_layout \begin_layout Subsection Behind the scene \end_layout \begin_layout Standard Even though there are installation programs that performs all these steps automatically,it is better to know what is really happening. Installing an operating system involves; \end_layout \begin_layout Itemize Partitioning of the hard disk \end_layout \begin_layout Itemize Formatting the partitions \end_layout \begin_layout Itemize Copying the operating system files \end_layout \begin_layout Itemize Installing a boot loader program \end_layout \begin_layout Standard The storage space of a hard disk drive can be divided into separate data areas, known as partitions. You can create primary partitions and extended partitions. Logical drives (secondary partitions can be created inside the extended partitions). On a disk, you can have up to 4 partitions, where one of them could be an extended partition. You can have many logical drives inside the extended partition. \end_layout \begin_layout Standard On a MSWindows system, the primary partition is called the C: drive. The logical drives inside the extended partition are named from D: onwards. GNU/Linux uses a different naming convention. The individual disks are named as /dev/sda , /dev/sdb etc. and the partitions inside them are named as /dev/sda1, /dev/sda2 etc. The numbering of secondary partitions inside the logical drive starts at /dev/sda5. (1 to 4 are reserved for primary and extended). Hard disk partitioning can be done using the fdisk program. The installation program also does this for you. \end_layout \begin_layout Standard The process of making a file system on a partition is called formatting. There are many different types of file systems. MSWindows use file systems like FAT32, NTFS etc. and GNU/Linux mostly uses file systems like ext3, ext4 etc. \end_layout \begin_layout Standard The operating system files are kept in directories named boot, sbin, bin, etc etc. The kernel that loads while booting the system is kept in /boot. The configuration files are kept in /etc. /sbin and /bin holds programs that implements many of the shell commands. Most of the application programs are kept in /usr/bin area. \end_layout \begin_layout Standard The boot loader program is the one provides the selection of OS to boot, when you power on the system. GRUB is the boot loader used by most of the GNU/Linux systems. \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "2" key "wikipedia" \end_inset http://en.wikipedia.org/wiki/List_of_curves \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "3" key "gap-system" \end_inset http://www.gap-system.org/~history/Curves/Curves.html \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "4" key "ellipse" \end_inset http://www.gap-system.org/~history/Curves/Ellipse.html \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "5" key "wolfram" \end_inset http://mathworld.wolfram.com/ \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "6" key "numpy examples" \end_inset http://www.scipy.org/Numpy_Example_List \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "7" key "scipy/doc" \end_inset http://docs.scipy.org/doc/ \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "8" key "Numerical Integration" \end_inset http://numericalmethods.eng.usf.edu/mws/gen/07int/index.html \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "9" key "fractals" \end_inset http://www.angelfire.com/art2/fractals/lesson2.htm \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "10" key "numerical recepies" \end_inset http://www.fizyka.umk.pl/nrbook/bookcpdf.html \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "11" key "mathcs.emory" \end_inset http://www.mathcs.emory.edu/ccs/ccs315/ccs315/ccs315.html \end_layout \begin_layout Bibliography \begin_inset LatexCommand bibitem label "12" key "Kiusalas" \end_inset Numerical Methods in Engineering with Python by Jaan Kiusalaas \end_layout \end_body \end_document pycode-browser-1.02/pybooksrc/pics/000077500000000000000000000000001254404601000173455ustar00rootroot00000000000000pycode-browser-1.02/pybooksrc/pics/archi.png000077500000000000000000002442771254404601000211640ustar00rootroot00000000000000PNG  IHDR XvpsBIT|d pHYsaa?i IDATx{\WtN)9 6sP˘l6f3299r#D!D|>]i1}}x܏s_]׻T9鈢(H t."""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R"""""R""*/W^033C5"88?~011F2ǝ:u ͚51 ,UDD$#""*Pt;v@E{ )ulPP;;;߿˗/{丬,xxxHMMŔ)Suz鈢(]i޽{֭[ 77o&x ={: //׮]Zѹsg^9 88ƍ˗ׯ_qm5uDD.!"r9w}ݒ޽;@\\.]ÇpuuE&MgϞΝ;8q222qF4iD}_]5 ""PTTT*|<-<<jժ̱-[,5gsE>}P(ШQ#9rDE_iQ8::p.Ο?HII)¢9,,,JjҤIx1F|"" DDT.ƍ͛71yd 11ƍý{pS>i/'"r9r$.\ ~zɪU0eΑZ<UM DDTnӦMCJJ 3g 55h߾= տݸqy""@xv؁O?իW t۶m %;wC҉Hp""*7ob޽pqqnܸ 'OqɱAAAx/>}:jժK.A___ƯBDDѣGC Ga022*s1k,nnnXd j׮-CDD)@Hm8ԆԆԆԆԆԆԆԆԆԆԆԆԆԆԆԦ),,D$%%!55HKKCFF233l塰(BX.ttt```###F033C͚5QV-XXXVVV6l3DD@|DDD <<QQQw␘dgg#?? \:::z%B:::%a$//b_WWիWG5`ii kkk888Ahܸ1CCÊHtħ/OQ'O"88ׯ_Gtt4`CCCԨQFѸqc888v%=ְJjǏX󒜜T{.~M055 5jVZK.֭TR;8y$^۷o#..P(5665LLLмys4oM6E-Ú ׯݻAll,bbbc:ZjY&ѬY3m\\\˩DDBDA w^;wHII `aah߾=GG*S-"""pI\t aaax!rrrPXXXr,--ѴiSt[oUDDBDA^6tuuaii͛sѣ:u9`9rHLLDzzzɰ.""c!"Bݻ燐$$$ kFfoch׮>-A+Ww^e`jj*wyDJo;233.]`ҥpvv<"" DT :t&LݻwNNN7oeH}vڅI&Ç7|˗/G߾}e:"4lقoA޽n:ԭ[W҈d#|8pA@:uh" 6L҈}DAҥKaii#F )) | q!֭}!-- Æ Ó'O0|pXZZbٲeAai%B3gbŊɁ!>s,ZkNB,:u*֭[|bܸq7oW#"`!"'b˖-(((|||0k,nƦ@4 O?!331b|}}riBSbժU(**5͛1c][t).\d/Ē%K#BD*BDo…?>rssallիWcĈrETرƍCRR0gL6Mx"X;,,,7@WW rrr>TdȐ!HLL!"kXXX`˖-rFD{@H9rL:"53fePPPubÆ իܥc!"q% >o߆ݱ}vj8 ]bرؼy3Ѽyslٲ;Q1>>GHHݻatϟGݺu 蕰dq|'HKKv܉.]]ӧ"99j–-[Я_?""-"RxtnnnAev튤$|Ά:txK#" BDj!ƍcǎHNNƌ3.^ ++ 'ODvvv 6`ٲek׮j_nƏ_MP@GGw >L'OM[p!ѣGǏ,"z!""۷o:u`ΝܹeUY066.7|kkk?'|ڵk?-f͚ޫLB/..^u֡e˖puu-i;rn݊m۶AGGT͖^zGu9077͛1`"" BD*>}ҥKٳw]V|lٲ-<<?#|}}aee:T VNNLLLJ͘1[gIۃggRaTܹsQTT:tT%Tb՘8q"ЪU+9sfffrU)%%%aǎQVM6={:ݾ}vw}W*\p͛7ϥdffbРA믿___;VH0R%''gϞz*LMM޽{]VGYf%mIIItw###.?3Znw}-..`oo/WYΑ#G|mǎcoQBDJzjL0 nnnصkw1M(z~ڵ011ydmz}ΝC`` .\X]rݖ*,,ĠA"b!lv777jyyyٳѱcG˩rrr9sCrՎ9OOOdgg7 c!\zT̩S`ݺur.;vAx뭷.E ǎcoQBD¹WjFJJ WSSSR@n޼$t޽\m1tPUAJDl׮]իWѣG0|_-[j@ ТERg B!OQZ )))pssիWQNڵKH BD& ???" },$"\ԭ[WrHF8z(}]F:t B///l۶*1vQƢ^z󃓓?~A.dVTTHDDD]۷/`ee^"M!XDUojժZjԨ;wСCezR^ 8p ƍCCC9sJǏcZjܡDЬY:<^*SE3g.\7jm69sVzw~sN8 lܸQ Ø1c`eeoo2m۶ FFF04436o\/44Yfر#<<}^zHNNoݻwcƌ7o^sFEEHiK\~;wFvv6ƌk*$'' cc*Wg}ΝCVTvA!11O M8(5y|ŊCVSNa…ppp@hh(LLLJ3%% h֬Y#DTaIII8wsU144T;u$ZXXW^}===E1))I%uVQOOO׮]kiQ J(ߑ#Gď?X2dqFQWWWnݪkիWo߾]6b{AQGGGUImDڎћHڴiggg 4'N曗~"\rEe?Ç!.^1cƨZnׯeK^uVːͨQpc?~Rϟ Ǐ ӑ^қ~~߾}QV-\pAuU\H 舘YP(ڵ+Ο? ҥK&uZnRSSKVE 'HK瑞x}""401 z=^;00>}z{qqqprrŠ+JVVom~ ?Skժb/<D/BFYYY +ײ[nڷo=zggg$$$o߾ rܪݻ@T?![B5VVRyyN^_~) }={DAAME)ܚnRDUMOȸ{))gn.FjE?}kl  /8;HL ǩekkѱBr}JRTnΜ9;w. _k׮Jƿ'O?'7n Qqi,Yo.\PfCB"b! 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It must be initialized with a root directory. """ def __init__( self, root_dir, filter_out_extensions = [] ): self.tmpfiles=[] # files to be deleted upon quit self.root_dir = root_dir self.filter_out_ext = filter_out_extensions ## The store will contain: ## 0. filename (str) ## 1. whether it's a directory (bool) ## 2. file size (int) ## 3. last modified (str) column_mapping = [ 0, 2, 3 ] # From user columns to model columns self.file_structure = Gtk.TreeStore(str, bool) self.populate_tree(root_dir, self.file_structure.get_iter_first()) uifile = abs_path_gui("gui.ui") wTree = Gtk.Builder() wTree.add_from_file(uifile) ## Create the treeview and link it to the model # self.w_treeview = wTree.get_widget("treeview") self.w_treeview = wTree.get_object("treeview") self.w_treeview.set_model(self.file_structure) self.srcView = GtkSource.View() self.srcBfr = self.srcView.get_buffer() srcLanguage = get_language_for_mime_type("text/x-python") #self.helpBfr = Gtk.TextBuffer() self.srcScrolledWindow = wTree.get_object("srcScrolledWindow") self.srcScrolledWindow.add(self.srcView) self.srcBfr.set_language(srcLanguage) self.srcBfr.set_highlight_syntax(True) self.srcView.set_buffer(self.srcBfr) self.srcView.set_show_line_numbers(True) context = self.srcView.get_pango_context() font = context.get_font_description() font.set_size(int(font.get_size() * 1.5)) self.srcView.modify_font(font) #self.srcView.set_editable(False) self.srcView.show() self.terminalScrolledWindow = wTree.get_object("terminalScrolledWindow") self.terminal_expander = wTree.get_object("terminalexpander") self.terminal = Vte.Terminal() self.terminalScrolledWindow.add(self.terminal) self.terminal.show() #self.helpView = wTree.get_widget("helpView") #self.helpView.set_buffer(self.helpBfr) self.srcBfr.set_text("#Python Code Browser: Select a python program from the left panel") self.btnExecute=wTree.get_object("btnExecute") self.btnSaveas=wTree.get_object("btnSaveas") self.tbtnExecute=wTree.get_object("tbtnExecute") self.tbtnSaveas=wTree.get_object("tbtnSaveas") dic={"on_frm_treeview_delete_event": self.quit, "quit_clicked": self.quit, "execute_clicked":self.open_file, "on_treeview_cursor_changed":self.disp_details, "saveas_clicked":self.save_as, "about_clicked":self.about} wTree.connect_signals(dic) ## Create the columns to view the contents self.columns = [ Gtk.TreeViewColumn(title) for title in ['Filename'] ] self.w_cell = Gtk.CellRendererText() self.w_cell.set_property("xalign",0) for i,column in enumerate(self.columns): self.w_treeview.append_column(column) column.set_property("resizable", True) column.pack_end(self.w_cell, True) column.add_attribute(self.w_cell, 'text', column_mapping[i]) ## Create a cell-renderer that displays a little directory or ## document icon depending on the file type self.w_cellpix = Gtk.CellRendererPixbuf() self.w_cellpix.set_property("xpad", 8) self.w_cellpix.set_property("xalign", 0) def pix_format_func(treeviewcolumn, cell, model, iter, dummyParam): if model.get(iter,1)[0]: cell.set_property("stock-id", Gtk.STOCK_DIRECTORY) else: cell.set_property("stock-id", Gtk.STOCK_ABOUT) self.columns[0].pack_start(self.w_cellpix, expand=True) self.columns[0].set_cell_data_func(self.w_cellpix, pix_format_func) def quit(self,*args): for f in self.tmpfiles: os.unlink(f) Gtk.main_quit() return def execute (self,src): """ Executes Python code with Python2; if the editor's content has been modified, takes its code from the editor's content. @param src : a path to the source file loaded into the editor """ cmd = "/usr/bin/python" tmpfile=None if self.srcBfr.get_modified()==True: tmpfile=NamedTemporaryFile(mode='w', suffix='.py', prefix='pycode-007-', delete=False) tmpfile.write(self.srcBfr.get_text(self.srcBfr.get_start_iter(), self.srcBfr.get_end_iter(), include_hidden_chars=False)) tmpfile.close() argv = [cmd, tmpfile.name] self.tmpfiles.append(tmpfile.name) # to delete the file later else: argv = [cmd, src] self.terminal.reset(True, True) self.terminal.grab_focus() print "about to launch", argv self.terminal.spawn_sync (pty_flags=Vte.PtyFlags.DEFAULT, working_directory='.', argv=argv, envv=[], spawn_flags=GLib.SpawnFlags.DO_NOT_REAP_CHILD, child_setup=None, child_setup_data=None,) print "done" self.terminal_expander.set_expanded(True) return def open_file(self,obj): model, parent_iter = self.w_treeview.get_selection().get_selected() pathname = self.get_pathname_from_iter(parent_iter) extn = os.path.splitext(pathname)[1] if extn == ".py": self.execute(pathname) def about(self,obj): abouttxt="Python Code Browser: Version 0.93\nCode: Vibeesh P., Vimal Joseph\nLicense: GNU GPL V3" #self.helpBfr.set_text(abouttxt) cmd = "echo" argv = [cmd, abouttxt] self.terminal.reset(True, True) self.terminal.fork_command(command=cmd,argv=argv) self.terminal_expander.set_expanded(True) def save_as(self,obj): dialog = Gtk.FileChooserDialog(title=None,action=Gtk.FILE_CHOOSER_ACTION_SAVE,buttons=(Gtk.STOCK_CANCEL,Gtk.RESPONSE_CANCEL,Gtk.STOCK_SAVE,Gtk.RESPONSE_OK)) model, parent_iter = self.w_treeview.get_selection().get_selected() fpath = self.get_pathname_from_iter(parent_iter) path,filename = os.path.split(fpath) dialog.set_current_name(filename) response = dialog.run() if response == Gtk.RESPONSE_OK: if self.srcBfr.get_modified()==True: f = open(dialog.get_filename(),"w") f.write(self.srcBfr.get_text(self.srcBfr.get_start_iter(), self.srcBfr.get_end_iter())) f.close() else: shutil.copy(fpath,dialog.get_filename()) elif response == Gtk.RESPONSE_CANCEL: print 'Closed, no files selected' dialog.destroy() def disp_details(self,view): model, parent_iter = view.get_selection().get_selected() path = self.get_pathname_from_iter(parent_iter) extn = os.path.splitext(path)[1] if extn == ".py": self.btnExecute.set_sensitive(True) self.btnSaveas.set_sensitive(True) self.tbtnExecute.set_sensitive(True) self.tbtnSaveas.set_sensitive(True) fname_without_extn = os.path.splitext(path)[0] desc_fname = fname_without_extn + ".py" if os.path.isfile(desc_fname) == True: fl = open(desc_fname) desc = fl.read() fl.close() hdesc = "Click on Execute to run this program\nSave as to save the program and modify it" #add the hdesc to status bar else: desc="#Python Code Browser: Select a python program from the left panel" hdesc="Select a python program from this directory" self.btnExecute.set_sensitive(False) self.btnSaveas.set_sensitive(False) self.tbtnExecute.set_sensitive(False) self.tbtnSaveas.set_sensitive(False) self.srcBfr.set_text(desc) #self.helpBfr.set_text(hdesc) self.srcBfr.set_modified(False) def get_pathname_from_iter( self, treeiter ): """Return a filesystem pathname from a tree in the path. This involves looking up the filenames at each step and joining them. """ m = self.file_structure if treeiter: # Not at root treepath = m.get_path(treeiter) filenames = [ ] while treeiter is not None: filenames.append(m.get_value(treeiter, 0)) treeiter = m.iter_parent(treeiter) filenames.reverse() else: filenames = [ ] return os.path.join(*[self.root_dir]+filenames) def populate_tree( self, dir, treeparent, visit_subdirectories = True ): """Given a location in the tree (given by treeparent), fill out the file information. Repopulation (i.e. calling a second time to update the tree) is not very well handled right nself.execute(pathname)ow. """ m = self.file_structure files = [] for f in os.listdir(dir): try: if f[0] != '.': if os.path.isdir(os.path.join(dir,f)): if len(os.listdir(os.path.join(dir,f))) > 0 and f!="gui": files.append(f) elif os.path.splitext(f)[1] in self.filter_out_ext: files.append(f) except OSError,e: print e files.sort() ## Stat each file andself.lbl_desc construct the row for f in files: row = [ f, False] fname = os.path.join(dir, f) try: filestat = os.stat(fname) row[1] = stat.S_ISDIR(filestat.st_mode) except OSError: pass m.append(treeparent, row) ## Populate subdirectories if required if visit_subdirectories and len(files) > 0: n = m.iter_n_children(treeparent) for i in range(n): child_iter = m.iter_nth_child(treeparent,i) if m.get_value(child_iter,1): # It's a subdirectory self.populate_tree(os.path.join(dir,files[i]), child_iter, visit_subdirectories) def set_double_click_callback( self, func ): """The callback is called with the following arguments: - filebrowser object - pathname of the file clicked (not necessarily absolute) - whether the file is a directory (bool """ def treeview_callback(treeview, path, view_column): file_iter = treeview.get_model().get_iter(path) isdir = self.file_structure.get_value(file_iter, 1) pathname = self.get_pathname_from_iter(file_iter) func(self, pathname, isdir) self.w_treeview.connect("row-activated", treeview_callback) ##### Standalone Running ################################################## if __name__ == "__main__": def my_callback(fb,pathname,isdir): extn = os.path.splitext(pathname)[1] if extn == ".py": fb.execute(pathname) fb = FileBrowser_pycode(os.path.join(abs_path(),'Code'),[".py"]) fb.set_double_click_callback(my_callback) Gtk.main()

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