Hey guys, I know that Simpson's rule has to be applied for even n rectangles, however in my assignment I am asked to calculate in 7 rectangles between a = 0 and b = 1. I have read forum links on this problem, but couldn't find the answer. Therefore, I am asking what I should change in my code to get a correct answer of integral = 0.333 for 7 rectangles (Currently I get 0.293). When I calculate for 12, 10, 8, etc with a = 0 and b = 1 everything is fine - I get integral = 0.333.
One way of deriving Simpson's rule is to fit a quadratic through 3 points and then integrate over a double interval of length 2h. If n is odd then you have one interval of length h left over. In this case, if you fit a quadratic through the last 3 points (b, b-h and b-2h) but only integrate it over the last h interval you get (after a lot of algebra) a contribution
Then I have to subtract off what your code already includes for this interval. The way that you have done the sums gives
The first f comes from half of the sum_even contribution and the second from the end of the interval.
If you subtract these (carefully) you will get the additive correction that I gave.
WARNING: this depends very much on how your original code treats the last interval, and different implementations will require different multipliers.
I have no idea whether this is the recommended technique - I simply hashed it out on the train on the way to work. However, it is still second-order and will give the exact result for a quadratic test function ... as you have here.
The reason I asked about your assignment is that I think it slightly more like that you might be required to do a check that n was even.
Why not bisect the last (uncounted) slice and apply Simpsons rule to those 2 subslices (3 values), and add the result to the rest?
After subtracting off what the code effectively had for this interval already it leads to the alternative additive correction if ( n % 2 != 0 ) result += ( h / 6 ) * ( -fk(b-h) + 4 * fk(b-0.5*h) - fk(b) );
and it produces the correct answer.
Depends whether this now constitutes 8 (unequal) intervals instead of 7, I guess.
Yeah, I'll post it in a second. It is similar to lastchance but a slightly different approach in that I take one slice off and put it back again as a new slice the same width as all the others, but it is separately split in two.