SICP 読み (87) 2.5.3 例: 記号代数
あと 4 頁がなかなか進まん。来週以降は対応可能な時間も減る。
問題 2.91
問題 2.88 の解をコピッて作成する方向で。まず試験か。div-poly は商と剰余なリストを返す、という事に。
("2-91"
(setup (lambda ()
(install-scheme-number-package)
(install-polynomial-package)))
("example"
(assert-equal '(polynomial x (((3 1) (1 1)) ((1 1) (0 -1))))
(div (make-polynomial 'x '((5 1) (0 -1)))
(make-polynomial 'x '((2 1) (0 -1)))))
)
("x^2 - 1/ x^2 - 1"
(assert-equal '(polynomial x (((0 1)) ()))
(div (make-polynomial 'x '((2 1) (0 -1)))
(make-polynomial 'x '((2 1) (0 -1)))))
)
("x^2 + 2x + 1 / x + 1"
(assert-equal '(polynomial x (((1 1) (0 1)) ()))
(div (make-polynomial 'x '((2 1) (1 2) (0 1)))
(make-polynomial 'x '((1 1) (0 1)))))
)
)なんか微妙。結果としては
'((polynomial x ((1 1) (0 1))) (polynomial x ()))
の方が多項式としては自然に見える。この方向で実装を検討してみる。
検討
処理としては
- 被除数の最高次の項を除数の最高次の項で割る。この結果が商の第一項。(make-term new-o new-c) すれば良い。
- この結果に除数を掛ける。
- 上の結果を被除数から引く。これと除数を div-term に再帰的に渡す。
- 除数の次数が被除数の次数を越えたら停止し、その被除数を剰余とする。
最初とケツのソレは提示されてますな。2 番目と 3 番目が_結果の残りを再帰的に (以下略)_な部分ですか。
で、rest-of-result には商と剰余のリストが入ってるハズなんで、完全な結果は
(list (cons (make-term new-o newc) (car rest-of-result)) (cadr rest-of-result))
で形成できるんかなぁ。
実装
上記を踏まえ、以下の実装を。require してる 2.5.1 は略。
(require "2.5.1")
(define (install-polynomial-package)
(define (adjoin-term term term-list)
(if (=zero? (coeff term))
term-list
(cons term 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 (make-poly variable term-list)
(cons variable 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)))
(define (add-poly p1 p2)
(if (same-variable? (variable p1) (variable p2))
(make-poly (variable p1)
(add-terms (term-list p1)
(term-list p2)))
(error "Polys not in same var --- ADD-POLY"
(list p1 p2))))
(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 (mul-poly p1 p2)
(if (same-variable? (variable p1) (variable p2))
(make-poly (variable p1)
(mul-terms (term-list p1)
(term-list p2)))
(error "Polys not in same var --- MUL-POLY"
(list p1 p2))))
(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 (neg-poly p)
(make-poly (variable p)
(neg-terms (term-list p))))
(define (neg-terms L)
(if (empty-termlist? L)
(the-empty-termlist)
(let ((t (first-term L)))
(adjoin-term
(make-term (order t) (- (coeff t)))
(neg-terms (rest-terms L)))))
)
(define (div-poly p1 p2)
(define (quotient solution)
(car solution)
)
(define (reminder solution)
(cadr solution)
)
(if (same-variable? (variable p1) (variable p2))
(let ((solution (div-terms (term-list p1) (term-list p2))))
(list (make-poly (variable p1)
(quotient solution))
(make-poly (variable p1)
(reminder solution))))
(error "Polys not in same var --- DIV-POLY"
(list p1 p2)))
)
(define (div-terms L1 L2)
(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
(add-terms
(neg-terms
(mul-term-by-all-terms
(make-term new-o new-c)
L2))
L1)
L2)))
(list (cons (make-term new-o new-c) (car rest-of-result))
(cadr rest-of-result))
))))))
(define (tag p) (attach-tag 'polynomial p))
(put 'div '(polynomial polynomial)
(lambda (p1 p2)
(let ((result (div-poly p1 p2)))
(list (tag (car result)) (tag (cadr result))))))
(put '=zero? '(polynomial)
(lambda (p) (empty-termlist? (term-list p))))
(put 'neg '(polynomial)
(lambda (p) (tag (neg-poly p))))
(put 'add '(polynomial polynomial)
(lambda (p1 p2) (tag (add-poly p1 p2))))
(put 'mul '(polynomial polynomial)
(lambda (p1 p2) (tag (mul-poly p1 p2))))
(put 'make 'polynomial
(lambda (var terms) (tag (make-poly var terms))))
'done)
(define (make-polynomial var terms)
((get 'make 'polynomial) var terms))
(define (neg x) (apply-generic 'neg x))で試験してみたんですが、多項式のソレをまた間違えてた。正しくは
'((polynomial x ((1 1) (0 1))) (polynomial x ()))
ぢゃなくて
'((polynomial x (1 1) (0 1)) (polynomial x ()))
だな。
あ、違う。こうだ。
'((polynomial x (1 1) (0 1)) (polynomial x))
一応試験は通ったんですが、put してなくて、な不具合あり。試験は以下。
("2-91"
(setup (lambda ()
(install-scheme-number-package)
(install-polynomial-package)))
("example"
(assert-equal '((polynomial x (3 1) (1 1))
(polynomial x (1 1) (0 -1)))
(div (make-polynomial 'x '((5 1) (0 -1)))
(make-polynomial 'x '((2 1) (0 -1)))))
)
("x^2 - 1/ x^2 - 1"
(assert-equal '((polynomial x (0 1))
(polynomial x))
(div (make-polynomial 'x '((2 1) (0 -1)))
(make-polynomial 'x '((2 1) (0 -1)))))
)
("x^2 + 2x + 1 / x + 1"
(assert-equal '((polynomial x (1 1) (0 1))
(polynomial x))
(div (make-polynomial 'x '((2 1) (1 2) (0 1)))
(make-polynomial 'x '((1 1) (0 1)))))
)
)
