sicp-exercise-1.29-1.31

Exercise 1.29

n should be even

(define (simpson-integral f a b m)
  (define n (* m 1.0))
  (define step (/ (- b a) n))
  (define (next x) (+ x (* 2 step)))
  (* step
     (/ 1.0 3.0)
     (+ (f a)
        (* 2
           (sum (next a) ; even-indexed elements
                (- b (* 2 step))
                f
                 next))
        (* 4
           (sum (+ a step) ; odd-indexed elements
                (- b step)
                f
                next))
        (f b))))

This approach is somewhat ugly, the sequence is divided into even-indexed and odd-indexed, since I don't find a universal pattern.

Exercise 1.30

(define (sum-iter a b f next)
  (define (iter a result)
    (if (> a b)
      result
      (iter (next a)
            (+ result (f a)))))
  (iter a 0))

Exercise 1.31

Recursive one

(define (product a b term next)
  (if (> a b)
    1
    (* (term a)
       (product (next a) b term next))))

Iterative one

(define (product a b term next)
  (define (iter a result)
    (if (> a b)
      result
      (iter (next a)
            (* result (term a)))))
  (iter a 1))

Use product to define factorial.

(define (factorial n)
  (product 1 n identity inc))

Compute PI with John Wallis's formula.

(define (wallis-term m)
  (/ (* (* 2.0 m)
        (+ (* 2.0 m) 2.0))
     (square (+ (* 2.0 m) 1.0))))

(define (wallis-formula n)
  (product 1.0 n wallis-term inc))

(* 4.0 (wallis-formula 10000))