SICP再開。がんばろー。結局ひげぽんさんに追い付けなかった。
無限ストリームです。
3.53
(define s (cons-stream 1 (add-streams s s)))
はどのような結果を返すか頭で考えろ、という問題。
(stream-take s 10) ; => (1 2 4 8 16 32 64 128 256 512)
3.54 (項ごとの積を計算するmul-streamsと階乗の列factorials)
(define (mul-streams s1 s2) (stream-map * s1 s2)) (define factorials (cons-stream 1 (mul-streams factorials (integers-starting-from 2))))
3.55
(define (partial-sums s)
(cons-stream (stream-car s)
(add-streams (stream-cdr s)
(partial-sums s))))
(stream-take (partial-sums integers) 10)
; => (1 3 6 10 15 21 28 36 45 55)
3.56 (Hammingの問題)
2,3,5以外の数を素因数に持たない数の列を求める。
(define (merge s1 s2)
(cond ((stream-null? s1) s2)
((stream-null? s2) s1)
(else
(let ((s1car (stream-car s1))
(s2car (stream-car s2)))
(cond ((< s1car s2car)
(cons-stream s1car (merge (stream-cdr s1) s2)))
((> s1car s2car)
(cons-stream s2car (merge s1 (stream-cdr s2))))
(else
(cons-stream s1car
(merge (stream-cdr s1)
(stream-cdr s2)))))))))
(define S (cons-stream 1 (merge (scale-stream S 2)
(merge (scale-stream S 3)
(scale-stream S 5)))))
(stream-take S 20)
; => (1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 27 30 32 36)
3.57 (メモ化する/しない場合の計算数)
fibsの例で調べる。
(define *count* 0) (define (%add-streams s1 s2) (stream-map (lambda (x y) (set! *count* (+ *count* 1)) (+ x y)) s1 s2)) (define fibs (cons-stream 0 (cons-stream 1 (%add-streams (stream-cdr fibs) fibs)))) (stream-take fibs 10) ; メモ化しているとき: *count* => 9 ; していないとき: *count* => 221
3.58
以下のコードは何を表すか。
(define (expand num den radix)
(cons-stream
(quotient (* num radix) den)
(expand (remainder (* num radix) den) den radix)))分数num/denのradix進数展開かな。
(stream-take (expand 2 7 7) 10) ; => (2 0 0 0 0 0 0 0 0 0) (stream-take (expand 2 7 10) 10) ; => (2 8 5 7 1 4 2 8 5 7)
3.59 (べき級数)
べき級数の係数の無限列でいろいろする。
b (ex など)
(define exp-series (cons-stream 1 (integrate-series exp-series))) (define cosine-series (cons-stream 1 (integrate-series (stream-map - sine-series)))) (define sine-series (cons-stream 0 (integrate-series cosine-series)))
3.60 (級数の積)
(define (mul-series s1 s2)
(cons-stream (* (stream-car s1) (stream-car s2))
(add-streams (scale-stream (stream-cdr s2) (stream-car s1))
(mul-series (stream-cdr s1) s2))))チェック。
(define sin^2 (mul-series sine-series sine-series)) (define cos^2 (mul-series cosine-series cosine-series)) (define sin^2+cos^2 (add-streams sin^2 cos^2)) (define (calc-series series x n) (apply + (stream-take (mul-streams series (geometric-series 1 x)) n))) (calc-series sin^2+cos^2 10 10) ; => 0.9999999999936162 (calc-series sin^2+cos^2 100 10) ; => 0.9999930666572112 (calc-series sin^2+cos^2 0 10) ; => 1.0 (calc-series sin^2+cos^2 -1 10) ; => 1.0
3.61 (級数の逆数)
(define (invert-unit-series s)
(cons-stream
1
(stream-map
-
(mul-series (stream-cdr s)
(invert-unit-series s)))))
3.62 (級数の商)
(define (div-series s1 s2)
(let ((constant (stream-car s2)))
(if (zero? constant)
(error "denominator series should begin with a nonzero constant term")
(scale-stream (mul-series s1 (invert-unit-series (scale-stream s2 (/ constant))))
(/ constant)))
))
(define tangent-series (div-series sine-series cosine-series))でうまくいくと思ってたんだけど、ちゃんと計算できてないみたい(ここで躓いてストップしてた)。invert-unit-seriesが悪いのかとも思うけど、よくわかんないのでパス。
