For a numerical methods class, I need to write a program to evaluate a definite integral with Simpson's composite rule. I already got this far (see below), but my answer is not correct. I am testing the program with f(x)=x, integrated over 0 to 1, for which the outcome should be 0.5. I get 0.78746... etc.
I know there is a Simpson's rule available in Scipy, but I really need to write it myself.
I suspect there is something wrong with the two loops. I tried "for i in range(1, n, 2)" and "for i in range(2, n-1, 2)" before, and this gave me a result of 0.41668333... etc.
I also tried "x += h" and I tried "x += i*h". The first gave me 0.3954, and the second option 7.9218.
# Write a program to evaluate a definite integral using Simpson's rule with
# n subdivisions
from math import *
from pylab import *
def simpson(f, a, b, n):
for i in range(1,n/2):
x += 2*h
k += 4*f(x)
for i in range(2,(n/2)-1):
x += 2*h
k += 2*f(x)
def function(x): return x
print simpson(function, 0.0, 1.0, 100)
You probably forget to initialize
x before the second loop, also, starting conditions and number of iterations are off. Here is the correct way:
def simpson(f, a, b, n): h=(b-a)/n k=0.0 x=a + h for i in range(1,n/2 + 1): k += 4*f(x) x += 2*h x = a + 2*h for i in range(1,n/2): k += 2*f(x) x += 2*h return (h/3)*(f(a)+f(b)+k)
Your mistakes are connected with the notion of a loop invariant. Not to get into details too much, it's generally easier to understand and debug cycles which advance at the end of a cycle, not at the beginning, here I moved the
x += 2 * h line to the end, which made it easy to verify where the summation starts. In your implementation it would be necessary to assign a weird
x = a - h for the first loop only to add
2 * h to it as the first line in the loop.