Simpsons Rule Function | Python Fiddle

from math import sin, pi

def Simpson(f, a, b, n):
    if a > b:
        print 'Incorrect bounds'
        return None
    if n%2 != 0: # also an 'if' because both tests are NOT
        # mutually exclusive
        print 'Invalid choice of n'
        return None
    else:
    	h = (b - a)/float(n) # need to cast 'n' as float in order to avoid
        # integer division
    	sum1 = 0
    	for i in range(1, n/2 + 1):
        	sum1 += f(a + (2*i - 1)*h)
    	sum1 *= 4
    	sum2 = 0
    	for i in range(1, n/2): # range must be ints: range() integer 
            #end argument expected, got float.
        	sum2 += f(a + 2*i*h)
    	sum2 *= 2
    	approx = (b - a)/(3.0*n)*(f(a) + f(b) + sum1 + sum2)
    	return approx

'''    
print 1.5*Simpson(sin(x)**3, 0, pi)  NOT VALID B/C WE NEED TO DEFINE SIN(X)**3
AS IT'S OWN FUNCTION
'''

def f(x):
    return 1.5*sin(x)**3

print "Simpson approximation of 1.5*sin(x)**3:"
for n in 2,6,12,100,500:
	print 'For n=%g:   approx=%f,   error~%E' % \
    (n, Simpson(f, 0, pi, n), 2 - Simpson(f, 0, pi, n))
    
'''
Sample run:
Simpson approximation of f(x) = 1.5*sin(x)**3:
For n=2:   approx=3.141593,   error~-1.141593E+00
For n=6:   approx=1.989172,   error~1.082830E-02
For n=12:   approx=1.999489,   error~5.107670E-04
For n=100:   approx=2.000000,   error~9.752365E-08
For n=500:   approx=2.000000,   error~1.558627E-10
'''

'''need to use polynomial of degree <=2 to verify Simpson
a good candidate is g(x) = 3x**2, which will be exactly 1 
with bounds [0,1]'''

def g(x):
    return (3*x**2)

print
print "Simpson approximation of g(x) = 3*x**2:"
for n in 2,6,12,100,500:
	print 'For n=%g:   approx=%f,   error~%E' % \
    (n, Simpson(g, 0, 1, n), 1 - Simpson(g, 0, 1, n))
    
'''
Sample run:
Simpson approximation of g(x) = 3*x**2:
For n=2:   approx=1.000000,   error~0.000000E+00
For n=6:   approx=1.000000,   error~0.000000E+00
For n=12:   approx=1.000000,   error~2.220446E-16
For n=100:   approx=1.000000,   error~0.000000E+00
For n=500:   approx=1.000000,   error~3.330669E-16
'''

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