SICP/ex-3_63-xx.scm

(load "util.scm")

(define (pi-summands n)
  (cons-stream (/ 1. n)
               (stream-map - (pi-summands (+ n 2)))))

(define pi-stream
  (scale-stream (partial-sums (pi-summands 1)) 4))

(define (euler-transform s)
  (let ((s0 (stream-ref s 0))           ; Sn-1
        (s1 (stream-ref s 1))           ; Sn
        (s2 (stream-ref s 2)))          ; Sn+1
    (cons-stream (- s2 (/ (square (- s2 s1))
                          (+ s0 (* -2 s1) s2)))
                 (euler-transform (stream-cdr s)))))

(define (make-tableau transform s)
  (cons-stream s
               (make-tableau transform
                             (transform s))))

(define (accelerated-sequence transform s)
  (stream-map stream-car
              (make-tableau transform s)))

; (display (take 5 pi-stream))
; (newline)
; (display (take 5 (euler-transform pi-stream)))
; (newline)
; (display (take 5 (accelerated-sequence euler-transform pi-stream)))
; (newline)

(display "\nex-3.63 - sqrt-stream\n")

(define (sqrt-improve guess x)
  (average guess (/ x guess)))

(define (sqrt-stream x)
  (cons-stream 1.0
               (stream-map (lambda (guess)
                             (sqrt-improve guess x))
                           (sqrt-stream x))))

(define (sqrt-stream x)
  (define guesses
    (cons-stream 1.0
                 (stream-map (lambda (guess)
                               (sqrt-improve guess x))
                             guesses)))
  guesses)

(display (stream-ref (sqrt-stream 2) 1000))
(newline)

; The first implementation of sqrt-stream computes each value of the stream
; only once. Louis' suggestion computes all previous values because of the
; recursive calls to sqrt-stream. If memoization was not used the two solutions
; would behave in the same way.

(display "\nex-3.64 - stream-limit\n")

(define (stream-limit stream tolerance)
  (if (< (abs (- (stream-car stream)
                 (stream-car (stream-cdr stream))))
         tolerance)
    (stream-car (stream-cdr stream))
    (stream-limit (stream-cdr stream) tolerance)))

(define (sqrt-tol x tolerance)
  (stream-limit (sqrt-stream x) tolerance))

(assert (< (abs (- 1.4142135623730951 (sqrt-tol 2 0.01)))
           0.01) #t)

(assert (< (abs (- 4.795831523312719 (sqrt-tol 23 0.001)))
           0.001) #t)

(display "\nex-3.65 - ln2\n")

(define (ln2-summands n)
  (cons-stream (/ 1. n)
               (stream-map - (ln2-summands (+ n 1)))))

(define ln2-stream
  (partial-sums (ln2-summands 1)))

; slow
(define (ln2-tol tolerance)
  (stream-limit ln2-stream tolerance))

; fast
(define (ln2-tol tolerance)
  (stream-limit (accelerated-sequence euler-transform ln2-stream) tolerance))

(assert (ln2-tol 0.00000000001)
        0.6931471805599445)

; The series converges slowly. Only with acceleration we get a good result in
; reasonable time.

(display "\nex-3.66\n")

; (display "\nex-3.67\n")