Sicp Exercise 1.29

Exercise 1.29. Simpson’s Rule is a more accurate method of numerical integration than the method illustrated above. Using Simpson’s Rule, the integral of a function f between a and b is approximated as

where h = (b – a)/n, for some even integer n, and yk = f(a + kh). (Increasing n increases the accuracy of the approximation.) Define a procedure that takes as arguments f, a, b, and n and returns the value of the integral, computed using Simpson’s Rule. Use your procedure to integrate cube between 0 and 1 (with n = 100 and n = 1000), and compare the results to those of the integral procedure shown above.

	(define (even? n)
	(= (remainder n 2) 0))
	(define (inc n) (+ n 1))
	(define (cube x) (* (* x x) x))
	(define
	(sum-iter-helper
	runningsum
	termfunction
	termvalue
	nextfunction
	upperbound)
	(if
	(> termvalue upperbound)
	runningsum
	(sum-iter-helper
	(+ runningsum (termfunction termvalue))
	termfunction
	(nextfunction termvalue)
	nextfunction
	upperbound)))
	(define (sum term a next b)
	(sum-iter-helper
	0
	term
	a
	next
	b))
	(define (simpson-rule f a b n)
	(define (find-yk k)
	(if (even? k)
	(*
	2
	(f (+ a (* k (/
	(- b a)
	n)))))
	(*
	4
	(f (+ a (* k (/
	(- b a)
	n)))))))
	(*
	1.0
	(*
	(/
	(/
	(- b a)
	n)
	3)
	(sum
	find-yk
	0
	inc
	n)))

view raw simpsons_rule.scm hosted with by GitHub