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Exercise 3.64 stream-limit.rkt
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Exercise 3.64 stream-limit.rkt
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#lang racket
; Exercise 3.64. Write a procedure stream-limit that takes as arguments a stream and a number
; (the tolerance). It should examine the stream until it finds two successive elements that differ
; in absolute value by less than the tolerance, and return the second of the two elements.
; Using this, we could compute square roots up to a given tolerance by
; (define (sqrt x tolerance)
; (stream-limit (sqrt-stream x) tolerance))
; S O L U T I O N
(define (sqrt x tolerance)
(stream-limit (sqrt-stream x) tolerance)
)
(define (stream-limit s t)
(let ((s0 (stream-ref s 0)) (s1 (stream-ref s 1)))
(if (< (abs (- s1 s0)) t)
s1
(stream-limit (stream-rest s) t)
)
)
)
(define (sqrt-stream x)
(define guesses
(stream-cons
1.0
(stream-map
(lambda (guess) (sqrt-improve guess x))
guesses
)
)
)
guesses
)
(define (sqrt-improve guess x)
(average guess (/ x guess))
)
(define (average x y) (/ (+ x y) 2))
(define (scale-stream stream factor)
(stream-map (lambda (x) (* x factor)) stream)
)
(define (stream-map proc . argstreams)
; (displayln "Entered stream-map")
(if (stream-empty? (car argstreams))
empty-stream
(stream-cons
(apply proc (map stream-first argstreams))
(apply stream-map (cons proc (map stream-rest argstreams)))
)
)
)
(define (stream-ref s n)
; (display "Entered stream-ref with n = ")
; (display n)
; (newline)
(if (= n 0)
(stream-first s)
(stream-ref (stream-rest s) (- n 1))
)
)
; This procedure displays a finite number of elements from the supplied stream
; as specified by 'count'
(define (display-stream-elements count s)
(if (= 0 count)
(begin
(newline)
'done
)
(begin
(newline)
(display (stream-first s))
(display-stream-elements (- count 1) (stream-rest s))
)
)
)
(define (display-stream s)
(stream-for-each display-line s)
)
(define (display-line x)
(newline)
(display x)
)
(define (div-series dividend-series divisor-series)
(cond
((= 0 (stream-first divisor-series))
(error "Denominator should not have a zero constant: " (stream-first divisor-series))
)
(else
(mul-series
dividend-series
(invert-unit-series (scale-stream divisor-series (/ 1 (stream-first divisor-series))))
)
)
)
)
(define (invert-unit-series s)
(stream-cons
1
(mul-series
(scale-stream (stream-rest s) -1)
(invert-unit-series s)
)
)
)
(define (mul-series s1 s2)
(stream-cons
; (a0 * b0) (The following is the constant term of the series resulting from
; the multiplication)
(* (stream-first s1) (stream-first s2))
; The following is the rest of the series starting with the x^1 term
(add-streams
; {a0 * (b1x + b2x^2 + b3x^3 + ...)} +
; {b0 * (a1x + a2x^2 + a3x^3 + ...)} +
(add-streams
(scale-stream (stream-rest s2) (stream-first s1))
(scale-stream (stream-rest s1) (stream-first s2))
)
; {(a1x + a2x^2 + a3x^3 + ...) * (b1x + b2x^2 + b3x^3 + ...)}
(stream-cons
; 0 needs to be prepended to this stream so that the first term of the stream
; is the x^1 term. Only then the outer add-streams will add like terms in the
; two series supplied to it
0
(mul-series (stream-rest s1) (stream-rest s2))
)
)
)
)
(define exp-series
(stream-cons 1 (integrate-series exp-series))
)
; the integral of negative sine is cosine
(define cosine-series
(stream-cons 1 (integrate-series (scale-stream sine-series -1)))
)
; the integral of cosine is sine
(define sine-series
(stream-cons 0 (integrate-series cosine-series))
)
(define tan-series
(div-series sine-series cosine-series)
)
(define (integrate-series s)
(div-streams s integers)
)
(define (add-streams s1 s2)
(stream-map + s1 s2)
)
(define (div-streams s1 s2)
(stream-map / s1 s2)
)
(define ones (stream-cons 1 ones))
(define integers (stream-cons 1 (add-streams ones integers)))
; Test Driver
(define (run-test return-type proc . args)
(define (print-item-list items first-time?)
(cond
((not (pair? items)) (void))
(else
(if (not first-time?)
(display ", ")
(void)
)
(print (car items))
(print-item-list (cdr items) false)
)
)
)
(display "Applying ")
(display proc)
(if (not (null? args))
(begin
(display " on: ")
(print-item-list args true)
)
(void)
)
(newline)
(let ((result (apply proc args)))
(if (not (eq? return-type 'none))
(display "Result: ")
(void)
)
(cond
((procedure? result) ((result 'print)))
; ((eq? return-type 'deque) (print-deque result))
((eq? return-type 'none) (void))
(else
(print result)
(newline)
)
)
)
(newline)
)
(define (execution-time proc . args)
(define start-time (current-milliseconds))
; (display start-time)
; (display " ")
(apply proc args)
(define end-time (current-milliseconds))
; (display end-time)
(display "Execution time of ")
(display proc)
(display ": ")
(- end-time start-time)
)
; Tests
; Test Results
Welcome to DrRacket, version 6.11 [3m].
Language: racket, with debugging; memory limit: 1024 MB.
> (display-stream-elements 10 (sqrt-stream 2))
1.0
1.5
1.4166666666666665
1.4142156862745097
1.4142135623746899
1.414213562373095
1.414213562373095
1.414213562373095
1.414213562373095
1.414213562373095
'done
> (sqrt 2 10)
1.5
> (sqrt 2 5)
1.5
> (sqrt 2 1)
1.5
> (sqrt 2 .6)
1.5
> (sqrt 2 .5)
1.4166666666666665
> (sqrt 2 .1)
1.4166666666666665
> (sqrt 2 .09)
1.4166666666666665
> (sqrt 2 .08)
1.4142156862745097
> (sqrt 2 .01)
1.4142156862745097
> (sqrt 2 .009)
1.4142156862745097
> (sqrt 2 .005)
1.4142156862745097
> (sqrt 2 .003)
1.4142156862745097
> (sqrt 2 .002)
1.4142135623746899
>