Interestingly, your code gives the expected result with exact arithmetic: > (simpson-integral cube 0 1 100) 1/4
and less of an error for a smaller number of steps: > (simpson-integral cube 0 1.0 10) 0.24999999999999994 Suggestion: Modify the code to print out the intermediate steps in the calculation and compare inexact arithmetic with exact in detail for the n = 100 / h = 0.01 case. Dan On Mon, Nov 4, 2013 at 7:14 AM, Ben Duan <yfe...@gmail.com> wrote: > I've asked the question on Stack > Overflow<http://stackoverflow.com/questions/19706893/implementation-of-simpsons-rule-sicp-exercise-1-29>[1]. > Óscar López gave me an > answer. But I still don't understand it. So I re-post it here: > > Following is my code for SICP exercise 1.29 > <http://mitpress.mit.edu/sicp/full-text/book/book-Z-H-12.html>[2]. The > exercise asks us to > implement Simpson's Rule using higher order procedure `sum`. It's supposed > to be > more accurate than the original `integral` procedure. But I don't know why > it's > not the case in my code: > > (define (simpson-integral f a b n) > (define h (/ (- b a) n)) > (define (next x) (+ x (* 2 h))) > (* (/ h 3) (+ (f a) > (* 4 (sum f (+ a h) next (- b h))) > (* 2 (sum f (+ a (* 2 h)) next (- b (* 2 h)))) > (f b)))) > > Some explanations of my code: As > > h/3 * (y_{0} + 4*y_{1} + 2*y_{2} + 4*y_{3} + 2*y_{4} + ... + 2*y_{n-2} > + 4*y_{n-1} + y_{n}) > > equals > > h/3 * (y_{0} > + 4 * (y_{1} + y_{3} + ... + y_{n-1}) > + 2 * (y_{2} + y_{4} + ... + y_{n-2}) > + y_{n}) > > I just use `sum` to compute `y_{1} + y_{3} + ... + y_{n-1}` and `y_{2} + > y_{4} + ... + y_{n-2}`. > > Complete code here: > > #lang racket > (define (cube x) (* x x x)) > (define (sum term a next b) > (if (> a b) > 0 > (+ (term a) > (sum term (next a) next b)))) > (define (integral f a b dx) > (define (add-dx x) (+ x dx)) > (* (sum f (+ a (/ dx 2.0)) add-dx b) > dx)) > (define (simpson-integral f a b n) > (define h (/ (- b a) n)) > (define (next x) (+ x (* 2 h))) > (* (/ h 3) (+ (f a) > (* 4 (sum f (+ a h) next (- b h))) > (* 2 (sum f (+ a (* 2 h)) next (- b (* 2 h)))) > (f b)))) > > Some tests(The exact value should be 0.25): > > > (integral cube 0 1 0.01) > 0.24998750000000042 > > (integral cube 0 1 0.001) > 0.249999875000001 > > > (simpson-integral cube 0 1.0 100) > 0.23078806666666699 > > (simpson-integral cube 0 1.0 1000) > 0.24800798800666748 > > (simpson-integral cube 0 1.0 10000) > 0.2499999999999509 > > [1] > http://stackoverflow.com/questions/19706893/implementation-of-simpsons-rule-sicp-exercise-1-29 > [2] http://mitpress.mit.edu/sicp/full-text/book/book-Z-H-12.html > > Thanks > > ____________________ > Racket Users list: > http://lists.racket-lang.org/users > > -- *Daniel Prager* Agile/Lean Coaching, Software Development and Leadership Twitter: @agilejitsu <https://twitter.com/agilejitsu> Blog: agile-jitsu.blogspot.com
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