SICP-2.3.2节练习
练习 2.56 - 2.58
练习 2.56：
Show how to extend the basic differentiator to handle more kinds of expressions. For instance, implement the differentiation rule n_1 n_2 --- = --- if and only if n_1 d_2 = n_2 d_1 d_1 d_2 by adding a new clause to the deriv program and defining appropriate procedures exponentiation?, base, exponent, and make-exponentiation. (You may use the symbol ** to denote exponentiation.) Build in the rules that anything raised to the power 0 is 1 and anything raised to the power 1 is the thing itself.
我的解答：
(define (variable? x)
(symbol? x))
(define (same-variable? v1 v2)
(and (variable? v1)
(variable? v2)
(eq? v1 v2)))
(define (=number? exp num)
(and (number? exp)
(= exp num)))
(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 (sum? x)
(and (pair? x)
(eq? (car x) '+)))
(define (addend s)
(cadr s))
(define (augend s)
(caddr s))
(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 (product? x)
(and (pair? x)
(eq? (car x) '*)))
(define (multiplier p)
(cadr p))
(define (multiplicand p)
(caddr p))
(define (make-exponentiation base exponent)
(cond ((=number? exponent 0) 1)
((=number? exponent 1) base)
(else
(list '** base exponent))))
(define (exponentitation? x)
(and (pair? x)
(eq? (car x) '**)))
(define (base e)
(cadr e))
(define (exponent e)
(caddr e))
(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))))
((exponentitation? exp)
(make-product (exponent exp)
(make-product (make-exponentiation (base exp)
(- (exponent exp) 1))
(deriv (base exp) var))))
(else
(error "unknown expression type -- DERIV" exp))))
练习 2.57：
Extend the differentiation program to handle sums and products of arbitrary numbers of (two or more) terms. Then the last example above could be expressed as (deriv '(* x y (+ x 3)) 'x) Try to do this by changing only the representation for sums and products, without changing the deriv procedure at all. For example, the addend of a sum would be the first term, and the augend would be the sum of the rest of the terms.
我的解答：
(define (variable? x)
(symbol? x))
(define (same-variable? v1 v2)
(and (variable? v1)
(variable? v2)
(eq? v1 v2)))
(define (=number? exp num)
(and (number? exp)
(= exp num)))
(define (make-sum . as)
(cond ((null? as) 0)
((=number? (car as) 0) (apply make-sum (cdr as)))
((null? (cdr as)) (car as))
((and (number? (car as)) (number? (cadr as)))
(apply make-sum (cons (+ (car as)
(cadr as))
(cddr as))))
(else
(cons '+ as))))
(define (sum? x)
(and (pair? x)
(eq? (car x) '+)))
(define (addend s)
(cadr s))
(define (augend s)
(apply make-sum (cddr s)))
(define (make-product . ms)
(cond ((null? ms) 1)
((=number? (car ms) 0) 0)
((=number? (car ms) 1)
(apply make-product (cdr ms)))
((null? (cdr ms)) (car ms))
((and (number? (car ms)) (number? (cadr ms)))
(apply make-product (cons (* (car ms)
(cadr ms))
(cddr ms))))
(else
(cons '* ms))))
(define (product? x)
(and (pair? x)
(eq? (car x) '*)))
(define (multiplier p)
(cadr p))
(define (multiplicand p)
(apply make-product (cddr p)))
(define (make-exponentiation base exponent)
(cond ((=number? exponent 0) 1)
((=number? exponent 1) base)
(else
(list '** base exponent))))
(define (exponentitation? x)
(and (pair? x)
(eq? (car x) '**)))
(define (base e)
(cadr e))
(define (exponent e)
(caddr e))
(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))))
((exponentitation? exp)
(make-product (exponent exp)
(make-product (make-exponentiation (base exp)
(- (exponent exp) 1))
(deriv (base exp) var))))
(else
(error "unknown expression type -- DERIV" exp))))
练习 2.58：
Suppose we want to modify the differentiation program so that it works with ordinary mathematical notation, in which + and * are infix rather than prefix operators. Since the differentiation program is defined in terms of abstract data, we can modify it to work with different representations of expressions solely by changing the predicates, selectors, and constructors that define the representation of the algebraic expressions on which the differentiator is to operate. Show how to do this in order to differentiate algebraic expressions presented in infix form, such as (x + (3 * (x + (y + 2)))). To simplify the task, assume that + and * always take two arguments and that expressions are fully parenthesized. The problem becomes substantially harder if we allow standard algebraic notation, such as (x + 3 * (x + y + 2)), which drops unnecessary parentheses and assumes that multiplication is done before addition. Can you design appropriate predicates, selectors, and constructors for this notation such that our derivative program still works?
我的解答：
a:
(define (variable? x)
(symbol? x))
(define (same-variable? v1 v2)
(and (variable? v1)
(variable? v2)
(eq? v1 v2)))
(define (=number? exp num)
(and (number? exp)
(= exp num)))
(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 (sum? x)
(and (pair? x)
(not (null? (cdr x)))
(eq? (cadr x) '+)))
(define (addend s)
(car s))
(define (augend s)
(caddr s))
(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 (product? x)
(and (pair? x)
(not (null? (cdr x)))
(eq? (cadr x) '*)))
(define (multiplier p)
(car p))
(define (multiplicand p)
(caddr p))
(define (make-exponentiation base exponent)
(cond ((=number? exponent 0) 1)
((=number? exponent 1) base)
(else
(list '** base exponent))))
(define (exponentitation? x)
(and (pair? x)
(eq? (car x) '**)))
(define (base e)
(cadr e))
(define (exponent e)
(caddr e))
(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))))
((exponentitation? exp)
(make-product (exponent exp)
(make-product (make-exponentiation (base exp)
(- (exponent exp) 1))
(deriv (base exp) var))))
(else
(error "unknown expression type -- DERIV" exp))))
b:
还不会
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