NoSyu의 주저리주저리 :: SICP Exercise 연습문제 2.91

이 문제는 다항식의 나눗셈을 구하는 div-poly를 구하는 문제입니다.

정확하게는 div-poly 안의 div-terms 프로시저를 구해야합니다.

잘 되는군요.^^

다항식의 나눗셈은 다음과 같은 절차를 밟습니다.

(SICP 272쪽)

나뉘는 다항식(피제수, dividend, 여기서는 x^5 - 1)의 가장 큰 차수를 가진 항에서

나누는 다항식(제수, divisor, 여기서는 x^2 - 1)의 가장 큰 차수를 가진 항으로 나눕니다.

그렇게 하여 나온 식을 몫에 붙입니다.

여기서는 x^5 / x^2이므로 x^3을 몫에 붙이는겁니다.

그런 다음 앞에서 구한 x^3에 나누는 식을 곱합니다.

그럼 x^5 - x^3이 됩니다.

이를 나뉘는 식에 빼면 나머지가 나옵니다.

즉, (x^5 - 1) - (x^5 - x^3) = x^3 - 1이 나머지가 됩니다.

이 나머지를 다시 나뉘는 식으로 잡습니다.

즉, 이제 x^5 - 1 대신 x^3 - 1을 나뉘는 식으로 한 후 처음으로 돌아가는겁니다.

그럼 몫에 붙이는 것은 x가 될 것이고, 나머지는 x - 1이 나옵니다.

그럼 다시 x - 1을 나뉘는 식으로 잡은 후 처음으로 돌아갑니다.

이처럼 반복하다가 나머지의 가장 높은 차수가 나누는 식의 가장 높은 차수보다 작다면

그것이 본 식의 나머지가 되는 것입니다.

여기서는 x - 1이 x^ - 1보다 차수가 낮으므로 x - 1이 나머지가 되겠네요.

그리고 앞의 반복된 수행에서 몫에 붙인다고 한 것을 모두 더하면 그것이 몫이 됩니다.

처음에 x^3, 그 다음에 x가 나왔으니 몫은 x^3 + x가 되겠네요.

다항식의 나눗셈을 어떻게 프로그램으로 만들까 고민한 적이 있었습니다.

하지만 그 결과를 얻지 못했는데 이번 기회에 확실하게 알고가네요.^^

이번에 덧셈, 뺄셈, 곱셈을 이미 구현하였기에 나눗셈은 쉽게 만들 수있었습니다.

그리고 뺄셈에서 교환법칙이 성립하지 않음을 깜박하고

이를 반영하지 않은 코드도 바로 잡을 수있었습니다.^^

참조

해럴드 애빌슨, 김재우 역, <컴퓨터 프로그램의 구조와 해석>, 인사이트, 2007, pp. 272

(define true (= 0 0))
(define false (= 0 1))
(define (square x) (* x x))
; put/get
; in ch2support.scm - MIT support
(define (assoc key records)
(cond ((null? records) false)
        ((equal? key (caar records)) (car records))
        (else (assoc key (cdr records)))))
(define (make-table)
(let ((local-table (list '*table*)))
    (define (lookup key-1 key-2)
      (let ((subtable (assoc key-1 (cdr local-table))))
        (if subtable
            (let ((record (assoc key-2 (cdr subtable))))
              (if record
                  (cdr record)
                  false))
            false)))
    (define (insert! key-1 key-2 value)
      (let ((subtable (assoc key-1 (cdr local-table))))
        (if subtable
            (let ((record (assoc key-2 (cdr subtable))))
              (if record
                  (set-cdr! record value)
                  (set-cdr! subtable
                            (cons (cons key-2 value)
                                  (cdr subtable)))))
            (set-cdr! local-table
                      (cons (list key-1
                                  (cons key-2 value))
                            (cdr local-table)))))
      'ok)
    (define (dispatch m)
      (cond ((eq? m 'lookup-proc) lookup)
            ((eq? m 'insert-proc!) insert!)
            (else (error "Unknown operation -- TABLE" m))))
    dispatch))
(define operation-table (make-table))
(define get (operation-table 'lookup-proc))
(define put (operation-table 'insert-proc!))

; apply-generic
(define (apply-generic op . args)
; 층수를 반환
(define (floor p)
    (cond ((equal? p 'integer) 1)
          ((equal? p 'rational) 2)
          ((equal? p 'real) 3)
          ((equal? p 'complex) 4)
          (else (error "No package " p))))
; 리스트의 최고층을 반환
(define (high-floor args-list)
    (define (iter result list)
      (if (null? list)
          result
          (if (< result (floor (type-tag (car list))))
              (iter (floor (type-tag (car list))) (cdr list))
              (iter result (cdr list)))))
    (iter 0 args-list))
; 리스트를 살펴 최고층이 아닌 경우 raise
(define (raise-list high-floor args-list)
    (if (null? args-list)
        null
        (if (< (floor (type-tag (car args-list))) high-floor)
            (cons (raise (car args-list))
                  (raise-list high-floor (cdr args-list)))
            (cons (car args-list)
                  (raise-list high-floor (cdr args-list))))))
; 리스트가 모두 같은 층인가?
(define (same-floor? args-list)
    (define (iter list)
      (let ((high-f (high-floor args-list)))
        (cond ((null? list) true)
              ((< (floor (type-tag (car args-list))) high-f) false)
              (else (iter (cdr list))))))
    (iter args-list))
; 같은 층을 만드는 것.
(define (make-same-floor-list list)
    (if (same-floor? list)
        list
        (make-same-floor-list (raise-list (high-floor list) list))))
; 기존의 것
(define (p-apply-generic args-list)
    (let ((type-tags (map type-tag args-list)))
      (let ((proc (get op type-tags)))
        (if proc
            (apply proc (map contents args-list))
            (p-apply-generic (make-same-floor-list args-list))))))
; 실행
(p-apply-generic args))

; polynomial 패키지
(define (install-polynomial-package)
; 프로시저
(define (make-polynomial-dense variable term-list)
    ((get 'make-polynomial-dense 'dense) variable term-list))
(define (make-polynomial-sparse variable term-list)
    ((get 'make-polynomial-sparse 'sparse) variable term-list))
(define (adjoin-term term term-list)
    ((get 'adjoin-term 'sparse) term term-list))
(define (variable p) (car p))
(define (term-list p) (cdr p))
(define (variable? x) (symbol? x))
(define (same-variable? v1 v2)
    (and (variable? v1) (variable? v2) (eq? v1 v2)))
; 덧셈
(define (add-terms L1 L2)
    (cond ((empty-termlist? L1) L2)
          ((empty-termlist? L2) L1)
          (else
           (let ((t1 (first-term L1)) (t2 (first-term L2)))
             (cond ((> (order t1) (order t2))
                    (adjoin-term
                     t1 (add-terms (rest-terms L1) L2)))
                   ((< (order t1) (order t2))
                    (adjoin-term
                     t2 (add-terms L1 (rest-terms L2))))
                   (else
                    (adjoin-term
                     (make-term (order t1)
                                (add (coeff t1) (coeff t2)))
                     (add-terms (rest-terms L1)
                                (rest-terms L2)))))))))
(define (add-poly p1 p2)
    (if (same-variable? (variable p1) (variable p2))
        (make-polynomial-sparse (variable p1)
                                (add-terms (term-list p1)
                                           (term-list p2)))
        (error "Polys not in same var -- ADD-POLY"
               (list p1 p2))))
(define (add a b) (+ a b))
; 곱셈
(define (mul-terms L1 L2)
    (if (empty-termlist? L1)
        (the-empty-termlist)
        (add-terms (mul-term-by-all-terms (first-term L1) L2)
                   (mul-terms (rest-terms L1) L2))))
(define (mul-term-by-all-terms t1 L)
    (if (empty-termlist? L)
        (the-empty-termlist)
        (let ((t2 (first-term L)))
          (adjoin-term
           (make-term (+ (order t1) (order t2))
                      (mul (coeff t1) (coeff t2)))
           (mul-term-by-all-terms t1 (rest-terms L))))))
(define (mul-poly p1 p2)
    (if (same-variable? (variable p1) (variable p2))
        (make-polynomial-sparse (variable p1)
                                (mul-terms (term-list p1)
                                           (term-list p2)))
        (error "Polys not in same var -- MUL-POLY"
               (list p1 p2))))
(define (mul a b) (* a b))
; exercise 2.88 - 뺄셈
(define (sub-poly p1 p2)
    (if (same-variable? (variable p1) (variable p2))
        (make-polynomial-sparse (variable p1)
                                (sub-terms (term-list p1)
                                           (term-list p2)))
        (error "Polys not in same var -- ADD-POLY"
               (list p1 p2))))
(define (sub-terms L1 L2)
    (cond ((empty-termlist? L1) L2)
          ((empty-termlist? L2) L1)
          (else
           (let ((t1 (first-term L1)) (t2 (first-term L2)))
             (cond ((> (order t1) (order t2))
                    (adjoin-term
                     t1 (sub-terms (rest-terms L1) L2)))
                   ((< (order t1) (order t2))
                    (adjoin-term
                     (make-term (order t2)
                                (* -1 (coeff t2)))
                     (sub-terms L1 (rest-terms L2))))
                   (else
                    (adjoin-term
                     (make-term (order t1)
                                (sub (coeff t1) (coeff t2)))
                     (sub-terms (rest-terms L1)
                                (rest-terms L2)))))))))
(define (sub a b) (- a b))
; exercise 2.91 - 나눗셈
(define (div-poly p1 p2)
    (if (same-variable? (variable p1) (variable p2))
        (make-polynomial-sparse (variable p1)
                                (div-terms (term-list p1)
                                           (term-list p2)))
        (error "Polys not in same var -- MUL-POLY"
               (list p1 p2))))
(define (div-terms L1 L2) ; L1 : 분자, L2 : 분모
    (if (empty-termlist? L1)
        (list (the-empty-termlist) (the-empty-termlist))
        (let ((t1 (first-term L1)) (t2 (first-term L2)))
          (if (> (order t2) (order t1))
              (list (the-empty-termlist) L1)
              (let ((new-c (div (coeff t1) (coeff t2)))
                    (new-o (- (order t1) (order t2))))
                (let ((rest-of-result
                       (div-terms
                        (sub-terms L1
                                   (mul-terms
                                    (list (list new-o new-c)) L2))
                        L2)
                       ))
                  (list (add-terms (list (make-term new-o new-c))
                                   (car rest-of-result))
                        (cadr rest-of-result))
                  ))))))
(define (div a b) (/ a b))
; 인터페이스
(define (tag p) (attach-tag 'polynomial p))
(put 'add-poly '(polynomial polynomial)
       (lambda (p1 p2) (tag (add-poly (cdr p1) (cdr p2)))))
(put 'mul-poly '(polynomial polynomial)
       (lambda (p1 p2) (tag (mul-poly (cdr p1) (cdr p2)))))
(put 'sub-poly '(polynomial polynomial)
       (lambda (p1 p2) (tag (sub-poly (cdr p1) (cdr p2)))))
(put 'mul-poly '(polynomial polynomial)
       (lambda (p1 p2) (tag (mul-poly (cdr p1) (cdr p2)))))
(put 'make-polynomial-dense 'polynomial
       (lambda (var terms) (tag (make-polynomial-dense var terms))))
(put 'make-polynomial-sparse 'polynomial
       (lambda (var terms) (tag (make-polynomial-sparse var terms))))
(put 'div-poly '(polynomial polynomial)
       (lambda (p1 p2) (tag (div-poly (cdr p1) (cdr p2)))))
'done)

; 빽빽한 다항식(dense polynomial system)
(define (install-polynomial-dense-package)
; 프로시저
(define (make-polynomial-dense variable term-list)
    (define (recv current-order t-list)
      (if (null? t-list)
          null
          (if (= (order (first-term t-list)) current-order)
              (cons (first-term t-list)
                    (recv (- current-order 1) (rest-terms t-list)))
              (cons (list current-order 0)
                    (recv (- current-order 1) t-list)))))
    (cons variable (recv (order (first-term term-list)) term-list)))
(define (adjoin-term term term-list)
    ; 계수가 0이든 아니든 cons로 묶어낸다.
    (cons term term-list))
; 인터페이스
(define (tag p) (attach-tag 'dense p))
(put 'make-polynomial-dense 'dense
       (lambda (var terms) (tag (make-polynomial-dense var terms))))
(put 'adjoin-term 'dense
       (lambda (term term-list) (adjoin-term term term-list)))
'done)

; 성긴 다항식(sparse polynomial system)
(define (install-polynomial-sparse-package)
; 프로시저
(define (make-polynomial-sparse variable term-list)
    (cons variable term-list))
(define (adjoin-term term term-list)
    (if (=zero? (coeff term))
        term-list
        (cons term term-list)))
; 인터페이스
(define (tag p) (attach-tag 'sparse p))
(put 'make-polynomial-sparse 'sparse
       (lambda (var terms) (tag (make-polynomial-sparse var terms))))
(put 'adjoin-term 'sparse
       (lambda (term term-list) (adjoin-term term term-list)))
'done)

; 정의
(define (make-polynomial-dense variable term-list)
((get 'make-polynomial-dense 'polynomial) variable term-list))
(define (make-polynomial-sparse variable term-list)
((get 'make-polynomial-sparse 'polynomial) variable term-list))
(define (the-empty-termlist) '())
(define (first-term term-list) (car term-list))
(define (rest-terms term-list) (cdr term-list))
(define (empty-termlist? term-list) (null? term-list))
(define (make-term order coeff) (list order coeff))
(define (order term) (car term))
(define (coeff term) (cadr term))
(define (add-poly a b)
(apply-generic 'add-poly a b))
(define (=zero? x) (= x 0))
(define (sub-poly p1 p2)
(apply-generic 'sub-poly p1 p2))
(define (mul-poly p1 p2)
(apply-generic 'mul-poly p1 p2))

; type-tag
(define (attach-tag type-tag contents)
(cons type-tag contents))
(define (type-tag datum)
(cond ((pair? datum) (car datum))
(else (error "Bad tagged datum -- TYPE-TAG" datum))))
(define (contents datum)
(cond ((pair? datum) (cdr datum))
(else (error "Bad tagged datum -- CONTENTS" datum))))

; answer
(define (div-poly p1 p2)
(apply-generic 'div-poly p1 p2))

; execute
(install-polynomial-package) (install-polynomial-dense-package) (install-polynomial-sparse-package)
(define p1 (make-polynomial-sparse 'x
(list (list 5 1) (list 0 -1))))
(define p2 (make-polynomial-dense 'x
(list (list 2 1) (list 1 0) (list 0 -1))))
p1 p2
(newline)
(div-poly p1 p2)

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