算术包实例：符号代数练习题

练习2.87 以下将对多项式进行算术包构建：

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;;通用操作
(define (add x y) (apply-generic 'add x y))
(define (mul x y) (apply-generic 'mul x y))
(define (=zero? x) (apply-generic '=zero? x))
;;多项式运算中涉及的算术操作包
;;数字部分
(define (install-scheme-number-package)
  (define (tag-args op)
    (lambda (x y) (attach-tag 'scheme-number (op x y))))
  (put 'add '(scheme-number scheme-number) (tag-args +))
  (put 'mul '(scheme-number scheme-number) (tag-args *))
  (put '=zero? '(scheme-number)
       (lambda (x) (= x 0))))
;;符号代数部分
(define (install-polynomial-package)
    (define (number->poly variable n)
        (make-poly variable (adjoin-term (make-term 0 n) (the-empty-termlist))))
    (define (tag x) (attach-tag 'polynomial' x))
    (define (put-op name op)
        (put 'name '(polynomial polynomial)
                (lambda (x y) (tag (op x y))))
        (put 'name '(polynomial scheme-number)
                (lambda (x y) (tag (op x (number->poly (variable x) y)))))
        (put 'name '(scheme-number polynomial)
                (lambda (x y) (tag (op (number->poly (variable y) x)) y))))
    (put-op 'add add-poly)
    (put-op 'mul mul-poly)
    (put '=zero? 'polynomial =zero-poly?)
    'done)
;;对符号代数中需要的函数进行说明
(define (polynomial? p)
    (and (pair? p) (eq? (car p) 'polynomial)))
(define (make-poly variable term-list)
        (cons 'polynomial (cons variable term-list)))
(define (variable? p) (symbol? p))
(define (same-variable? p1 p2)
    (and (variable? p1) (variable? p2)(eq? p1 p2)))
(define (variable p)(cadr p))
(define (term-list p)(cddr p))
(define (add-poly p1 p2)
    (if (same-variable? (variable p1) (variable p2))
        (make-poly (variable p1)
                    (add-terme (term-list p1)
                               (term-list p2)))
        (error "Polys not in same var -- ADD" (list p1 p2))))
(define (mul-poly p1 p2)
    (if (same-variable? (variable p1) (variable p2))
        (make-poly (variable p1)
                    (mul-terme (term-list p1)
                               (term-list p2)))
        (error "Polys not in same var -- MUL" (list p1 p2))))
(define (add-terms L1 L2)
    (cond ((empty-termlist? L1) L2)
          ((empty-termlist? L2) L1)
          (else
            (let ((t1 (first-term L1))
                  (t2 (firet-term L2)))
                (cond ((> (order t1) (order t2))
                        (adjoin-term t1 (add-terms (rest-terms L1) L2)))
                      ((> (order t2) (order t1))
                        (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-terms L1 L2)
    (if (empty-termlist? L1)
        (the-empty-termlist)
        (add-terms (mul-terms-by-all-terms (first-term L1) L2)
                   (mul-terms (rest-terms L1) L2))))
(define (mul-terms-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-terms-by-all-terms t1 (rest-terms L))))))
(define (adjoin-term term term-list)
    (if (=zero? (coeff term))
        term-list
        (cons term term-list)))
;;零的判断
(define (=zero-poly? poly)
    (define (coeff-all-zero? term-list)
        (if (empty-termlist? term-list) #t
            (if (=zero? (coeff (first-term term-list)))
                (coeff-all-zero? (rest-of-terms term-list))
                #f)))
    (coeff-all-zero? (term-list poly)))
(define (first-term term-list) (car term-list))
(define (rest-of-terms term-list) (cdr term-list))
(define (make-term oredr coeff)(cond oredr coeff))
(define (order term)(car term))
(define (coeff term)(cadr term))
(define (empty-termlist? term-list)(null? term-list))

练习2.88 扩充算术包，加上多项式的减法。

点击查看代码

;;整数包中添加：
(put 'neg '(scheme-number) (lambda (x) (tag (- x))))
;;多项式中添加：
(put-op 'sub sub-poly)
(put 'neg '(polynomial) (lambda (x) (tag (neg-poly x))))
;;定义减法：
(define (sub-poly p1 p2)
  (add p1 (neg p2)))
;;通过一个通用的取负操作实现减法
(define (neg-poly p)
    (make-poly (variable p)
                (neg-term (term-list p))))
(define (neg-term L)
    (if (empty-termlist? L)
        (the-empty-termlist)
        (let ((t (first-term L)))
            (adjoin-term (make-term (order t) (neg (coeff t)))
                (neg-term (rest-of-terms L))))))