1 (define (memo-proc proc)
2 (let ((already-run? false) (result false))
6 (begin (set! already-run? true)
10 (define-syntax mydelay
11 (rsc-macro-transformer
14 `(memo-proc (lambda () ,exp)))))
16 (apply xfmr (cdr e))))))
18 (define (myforce delayed-object)
21 (define-syntax cons-stream
22 (rsc-macro-transformer
23 (let ((xfmr (lambda (x y) `(cons ,x (mydelay ,y)))))
25 (apply xfmr (cdr e))))))
27 (define (stream-car s)
29 (define (stream-cdr s)
31 (define stream-null? null?)
32 (define the-empty-stream '())
34 (define (integers-starting-from n)
35 (cons-stream n (integers-starting-from (+ n 1))))
37 (define (stream-ref s n)
40 (stream-ref (stream-cdr s) (- n 1))))
41 (define (stream-map proc . argstreams)
42 (if (stream-null? (car argstreams))
45 (apply proc (map stream-car argstreams))
46 (apply stream-map (cons proc (map stream-cdr argstreams))))))
47 (define (stream-for-each proc s)
50 (begin (proc (stream-car s))
51 (stream-for-each proc (stream-cdr s)))))
53 (define (stream-enumerate-interval low high)
58 (stream-enumerate-interval (+ low 1) high))))
59 (define (stream-filter pred s)
62 (let ((scar (stream-car s)))
64 (cons-stream scar (stream-filter pred (stream-cdr s)))
65 (stream-filter pred (stream-cdr s))))))
67 (define (display-stream s)
68 (stream-for-each display-line s))
69 (define (display-line x)
73 (define (test-case actual expected)
78 (display "Expected: ")
82 (define (integers-starting-from n)
83 (cons-stream n (integers-starting-from (+ n 1))))
84 (define integers (integers-starting-from 1))
86 (define (divisible? x y) (= (remainder x y) 0))
88 (stream-filter (lambda (x) (not (divisible? x 7)))
92 (cons-stream a (fibgen b (+ a b))))
93 (define fibs (fibgen 0 1))
100 (not (divisible? x (stream-car s))))
103 ;; (define primes (sieve (integers-starting-from 2)))
104 ;; (test-case (stream-ref primes 25) 101)
106 (define ones (cons-stream 1 ones))
107 (define (add-streams s1 s2)
108 (stream-map + s1 s2))
109 (define integers (cons-stream 1 (add-streams ones integers)))
110 ;; (test-case (stream-ref integers 15) 16)
115 (add-streams (stream-cdr fibs)
118 (define (scale-stream stream factor)
119 (stream-map (lambda (x)
122 (define double (cons-stream 1 (scale-stream double 2)))
127 (stream-filter prime? (integers-starting-from 3))))
130 (cond ((> (square (stream-car ps)) n) true)
131 ((divisible? n (stream-car ps)) false)
132 (else (iter (stream-cdr ps)))))
135 (define (mul-streams s1 s2)
136 (stream-map * s1 s2))
138 (define (partial-sums s)
140 (cons-stream (stream-car s)
145 (define (merge s1 s2)
146 (cond ((stream-null? s1) s2)
147 ((stream-null? s2) s1)
149 (let ((s1car (stream-car s1))
150 (s2car (stream-car s2)))
151 (cond ((< s1car s2car)
154 (merge (stream-cdr s1) s2)))
158 (merge s1 (stream-cdr s2))))
162 (merge (stream-cdr s1) (stream-cdr s2)))))))))
164 (define (test-stream-list stream list)
167 (begin (display "A: ")
168 (display (stream-car stream))
173 (test-stream-list (stream-cdr stream) (cdr list)))))
175 ;; Exercise 3.59. In section 2.5.3 we saw how to implement a polynomial arithmetic system representing polynomials as lists of terms. In a similar way, we can work with power series, such as
177 ;; represented as infinite streams. We will represent the series a0 + a1 x + a2 x2 + a3 x3 + ··· as the stream whose elements are the coefficients a0, a1, a2, a3, ....
179 ;; a. The integral of the series a0 + a1 x + a2 x2 + a3 x3 + ··· is the series
181 ;; where c is any constant. Define a procedure integrate-series that takes as input a stream a0, a1, a2, ... representing a power series and returns the stream a0, (1/2)a1, (1/3)a2, ... of coefficients of the non-constant terms of the integral of the series. (Since the result has no constant term, it doesn't represent a power series; when we use integrate-series, we will cons on the appropriate constant.)
183 (define (integrate-series a)
184 (stream-map / a integers))
186 ;; b. The function x ex is its own derivative. This implies that ex and the integral of ex are the same series, except for the constant term, which is e0 = 1. Accordingly, we can generate the series for ex as
189 (cons-stream 1 (integrate-series exp-series)))
191 ;; Show how to generate the series for sine and cosine, starting from the facts that the derivative of sine is cosine and the derivative of cosine is the negative of sine:
193 (define cosine-series
196 (integrate-series (stream-map - sine-series))))
200 (integrate-series cosine-series)))
202 Exercise 3.60. With power series represented as streams of coefficients as in exercise 3.59, adding series is implemented by add-streams. Complete the definition of the following procedure for multiplying series:
204 (define (mul-series s1 s2)
205 (cons-stream <??> (add-streams <??> <??>)))
207 You can test your procedure by verifying that sin2 x + cos2 x = 1, using the series from exercise 3.59.
