流（3.5.1~3.5.2）

;流实现的行为方式
;delay和force的实现
(define (memo-proc proc)
(let ((already-run? false) (result false))
(lambda ()
(if already-run?
result
(begin
(set! already-run? true)
(set! result (proc))
result)))))
;(delay exp)应该等价于下面这个式子,但不能用define,目前不知道该如何绑定
(memo-proc (lambda () exp))
(define (test-func)
(define (fib n)
(if (< n 2)
n
(+ (fib (- n 1)) (fib (- n 2)))))
(fib 30))
(define promise (memo-proc (lambda () (test-func))))
(define (my-force delayed-project)
(delayed-project))
;分别求值下面两个表达式，比较速度
(my-force promise)
(my-force promise)
;3.50
(define (my-stream-map proc . argstreams)
(if (empty-stream? (car argstreams))
the-empty-stream
(cons-stream
(apply proc (map stream-car argstreams))
(apply my-stream-map
(cons proc
(map stream-cdr argstreams))))))
(define a (cons-stream 1 (cons-stream 2 (cons-stream 3 the-empty-stream))))
(define b (cons-stream 1 (cons-stream 2 (cons-stream 3 the-empty-stream))))
(define c (cons-stream 1 (cons-stream 2 (cons-stream 3 the-empty-stream))))
(define (display-stream stream)
(if (empty-stream? stream)
the-empty-stream
(cons (stream-car stream)
(display-stream (stream-cdr stream)))))
(display-stream (my-stream-map + a b c))
;3.51
(define (stream-enumerate-interval low high)
(if (> low high)
the-empty-stream
(cons-stream low
(stream-enumerate-interval (+ 1 low) high))))
(define (display-line x)
(newline)
(display x))
(define (show x)
(display-line x)
x)
(define x (stream-map show (stream-enumerate-interval 0 10)))
(stream-ref x 5)
(stream-ref x 7)
;3.52
(define (stream-filter pred? stream)
(cond ((empty-stream? stream)
the-empty-stream)
((pred? (stream-car stream))
(cons-stream (stream-car stream)
(stream-filter pred? (stream-cdr stream))))
(else (stream-filter pred? (stream-cdr stream)))))
(define sum 0)
(define (accum x)
(set! sum (+ x sum))
sum)
(define seq (stream-map accum (stream-enumerate-interval 1 20)))
(define y (stream-filter even? seq))
(define z (stream-filter (lambda (x) (= (remainder x 5) 0))
seq))
(stream-ref y 7)
(display-stream z)
;无穷流
(define (integers-starting-from n)
(cons-stream n
(integers-starting-from (+ n 1))))
(define integers (integers-starting-from 1))
(define no-sevens
(stream-filter
(lambda (x) (not (= (remainder x 7) 0))) integers))
(define (fibgen a b)
(cons-stream a (fibgen b (+ a b))))
(define (divsible? x y)
(= (remainder x y) 0))
(define (sieve stream)
(cons-stream
(stream-car stream)
(sieve
(stream-filter
(lambda (x)
(not (divsible? x (stream-car stream))))
(stream-cdr stream)))))
(define primes (sieve (integers-starting-from 2)))
;隐式地定义流
(define ones (cons-stream 1 ones))
(define (add-streams s1 s2)
(stream-map + s1 s2))
(define (scale-stream stream factor)
(stream-map (lambda (x) (* x factor)) stream))
(define double (cons-stream 1  (scale-stream double 2)))
(define primes (cons-stream 2
(stream-filter prime?
(integers-starting-from
3))))
(define (prime? x)
(define (iter ps)
(let ((p (stream-car ps)))
(cond ((> (square p) x) true)
((= (remainder x p) 0) false)
(else (iter (stream-cdr ps))))))
(iter primes))
;3.53
(define s (cons-stream 1 (add-streams s s)))
(stream-ref s 8)
;3.54
(define (mul-streams s1 s2)
(cons-stream
(* (stream-car s1)
(stream-car s2))
(mul-streams
(stream-cdr s1)
(stream-cdr s2))))
(define factorials
(cons-stream 1 (mul-streams factorials
(integers-starting-from 2))))
;3.55
(define (partial-sums s)
(define the-partial-sums
(cons-stream (stream-car s)
(add-streams (stream-cdr s) the-partial-sums)))
the-partial-sums)
;nice and elegant,相当于从相反的方向构建
(define (partial-sums-2 s)
(define the-partial-sums
(add-streams s (cons-stream 0 the-partial-sums)))
the-partial-sums)
(define z (partial-sums-2 integers))
;3.56
(define (merge s1 s2)
(cond ((empty-stream? s1) s2)
((empty-stream? 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 s1car
(merge (stream-cdr s1)
(stream-cdr s2))))
(else
(cons-stream s2car
(merge s1 (stream-cdr s2)))))))))
(define s (cons-stream 1 (merge (scale-stream s 2)
(merge (scale-stream s 3)
(scale-stream s 5)))))
;3.58这个过程求的是radix进制下(num/den)的小数部分,需假设num<den
(define (expand num den radix)
(cons-stream
(quotient (* num radix) den)
(expand (remainder (* num radix) den) den radix)))
(define z (expand 1 7 10))
(define (show-n s n)
(if (= n 0)
'()
(cons (stream-car s)
(show-n (stream-cdr s) (- n 1)))))
(show-n z 6)
(show-n z 12)
;3.59
;;a
(define (integrate-series s)
(stream-map / s integers))
(define exp-series
(cons-stream 1 (integrate-series exp-series)))
(show-n exp-series 8)
;;b
(define cosine-series
(cons-stream 1
(scale-stream
(integrate-series sine-series)
-1)))
(define sine-series
(cons-stream 0
(integrate-series cosine-series)))
(show-n cosine-series 9)
(show-n sine-series 9)
;3.60
(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))))
(define (mul-series-2 s1 s2);o(n^2)
(add-streams
(scale-stream s2 (stream-car s1))
(cons-stream
0
(mul-series-2 (stream-cdr s1) s2))))

(define sine*cosine (mul-series sine-series cosine-series))
(show-n sine*cosine 10)
;3.61
(define (1/S s)
(define the-1/S
(cons-stream 1
(mul-series (scale-stream (stream-cdr s) -1)
the-1/S)))
the-1/S)
;warning:下面这个版本直接用递归，由于每次调用1/S时都会产生一个新的流，所以在计算过程中，记忆功能实际上并没有用到，但是算过一次后，两者的表现是一样的，读者可以分别运行两个版本多次比较差别
(define (1/S s)
(cons-stream 1
(mul-series (scale-stream (stream-cdr s) -1)
(1/S s))))
;3.62
(define (div-series s1 s2)
(if (= (stream-car s2) 0)
(error "div-series: divided by zero!")
(mul-series s1 (1/S s2))))
(define tan-series (div-series sine-series cosine-series))
(show-n tan-series 9)