Geraldine - 2 months ago 20

Python Question

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):

h=(b-a)/n

k=0.0

x=a

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)

return (h/3)*(f(a)+f(b)+k)

def function(x): return x

print simpson(function, 0.0, 1.0, 100)

Answer

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.