乐正

Actions speak louder than words.

Sicp-ex2-58

问题

假定我们希望修改求导程序,是它能用于常规数学公式,其中+和*采用的是中缀运算符而 不是前缀。由于求导程序是基于抽象数据定义的,要修改它,使之能用于另一种不同的表达 式表示,我们只需要换一套工作在新的,求导程序需要使用的代数表示式的表示形式上的谓 词、选择函数和构造函数。

a) 请说明怎样做出这些过程,以便完成在中缀表示形式(例如 (x + (3 * (x + (y + 2)))) )上的代数表达式求导。为了简化有关工作,现在可以假定+和*总是取两个参数,而且表达 式中已经加上了所有的括号。

b) 如果允许标准的代数写法,例如(x + 3 * (x + y + 2)),问题就会变得更困难许多。 在这种表达式里可能不写不必要的括号,并要假定乘法应该在加法之前完成。你还能为这种 表示方式设计好适当的谓词、选择函数和构造函数,使我们的求导程序仍能完成工作吗?

解答

练习2.58 (ex2-58.scm) download
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;; 求一个表达式的导数形式
(define (deriv exp var)
  (cond ((number? exp) 0)
        ((variable? exp)
         (if (same-variable? exp var) 1 0))
        ((sum? exp)
         (make-sum (deriv (addend exp) var)
                   (deriv (augend exp) var)))
        ((product? exp)
         (make-sum
           (make-product (multiplier exp)
                         (deriv (multiplicand exp) var))
           (make-product (deriv (multiplier exp) var)
                         (multiplicand exp))))
        ((exponentiation? exp)
         (make-product
           (make-product (exponent exp)
                         (make-exponentiation (base exp) (- (exponent exp) 1)))
           (deriv (base exp) var)))
        (else
          (error "unknown expression type -- DERIV" exp))))

(define (variable? x) (symbol? x))

(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))

(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        (else (list a1 '+ a2))))

(define (=number? exp num)
  (and (number? exp) (= exp num)))

(define (make-product m1 m2)
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        ((and (number? m1) (number? m2)) (* m1 m2))
        (else (list m1 '* m2))))

(define (sum? x)
  (and (pair? x) (eq? (cadr x) '+)))

(define (addend s) (car s))

(define (augend s) (caddr s))

(define (product? x)
  (and (pair? x) (eq? (cadr x) '*)))

(define (multiplier p) (car p))

(define (multiplicand p) (caddr p))

(define (exponentiation? x)
  (and (pair? x) (eq? (cadr x) '**)))

(define (base exp) (car exp))

(define (exponent exp) (caddr exp))

(define (make-exponentiation b e)
  (cond ((=number? e 0) 1)
        ((=number? e 1) b)
        (else (list b '** e))))

测试

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;; (load sicp/ex2-58.scm)

(make-sum 'x 'y)
;Value 19: (x + y)

(make-product 'x 'y)
;Value 20: (x * y)

(make-exponentiation 'x 4)
;Value 21: (x ** 4)

(deriv '(x + (3 * (x + (y + 2)))) 'x)
;Value: 4

(deriv '(x * y) 'x)
;Value: y

(deriv '((x * y) * (x + 3)) 'x)
;Value 22: ((x * y) + (y * (x + 3)))

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