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Exercise 3.61 invert-unit-series.rkt
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Exercise 3.61 invert-unit-series.rkt
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#lang racket
; Exercise 3.61. Let S be a power series (exercise 3.59) whose constant term is 1. Suppose we want to find the
; power series 1/S, that is, the series X such that S · X = 1. Write S = 1 + Sr where Sr is the part of S after
; the constant term. Then we can solve for X as follows:
; S . X = 1
; (1 + Sr) . X = 1
; X + Sr . X = 1
; X = 1 - Sr . X
; In other words, X is the power series whose constant term is 1 and whose higher-order terms are given by
; the negative of Sr times X. Use this idea to write a procedure invert-unit-series that computes 1/S for a
; power series S with constant term 1. You will need to use mul-series from exercise 3.60.
; S O L U T I O N
(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 (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)))
(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 (scale-stream stream factor)
(stream-map (lambda (x) (* x factor)) stream)
)
(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)
)
; 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
; See tests in SICP exercise 3.62