2.91
; A univariate polynomial can be divided by another one to produce a polynomial
; quotient and a polynomial remainder. For example,
;
; $\frac{x^5 - 1}{x^2 - 1} = x^3 + x$, remainder $x-1$
;
; Division can be performed via long division. That is, divide the highest-order
; term of the dividend by the highest-order term of the divisor. The result is the
; first term of the quotient. Next, multiply the result by the divisor, subtract
; that from the dividend, and produce the rest of the answer by recursively dividing
; the difference by the divisor. Stop when the order of the divisor exceeds the order
; of the dividend and declare the dividend to be the remainder. Also, if the dividend
; ever becomes zero, return zero as both quotient and remainder.
;
; We can design a div-poly procedure on the model of add- poly and mul-poly. The
; procedure checks to see if the two polys have the same variable. If so, div-poly
; strips off the variable and passes the problem to div-terms, which performs the
; division operation on term lists. div-poly finally reattaches the variable to the
; result supplied by div-terms. It is convenient to design div-terms to compute both
; the quotient and the remainder of a division. div-terms can take two term lists as
; arguments and return a list of the quotient term list and the remainder term list.
;
; Complete the following definition of div-terms by filling in the missing expressions.
; Use this to implement div-poly, which takes two polys as arguments and returns a
; list of the quotient and remainder polys.
; Add the following to `polynomial` package
(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
L1
(negate-terms
(mul-term-by-all-terms
(make-term new-o new-c)
L2)))
L2)))
(cons
(adjoin-term
(make-term new-o new-c)
(car rest-of-result))
(cdr rest-of-result))))))))
; Redefining the polynomial package
(define (install-polynomial-package)
;; internal procedures
;; representation of poly
(define (make-poly variable term-list) (cons variable term-list))
(define (variable p) (car p))
(define (term-list p) (cdr p))
; ⟨procedures same-variable? and variable? from section 2.3.2⟩
(define (variable? x) (symbol? x))
(define (same-variable? v1 v2)
(and (variable? v1) (variable? v2) (eq? v1 v2)))
;; representation of terms and term lists
;; ⟨procedures adjoin-term . . . coeff from text below⟩
;; Add
(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)))))))))
;; Multiply
(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) 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 (poly-zero? term-list)
(if (empty-termlist? term-list)
#t
(and (=zero? (coeff (first-term term-list)))
(poly-zero? (rest-terms term-list)))))
;; Subtraction
(define (negate-terms term-list)
(if (empty-termlist? term-list)
the-empty-termlist
(adjoin-term
(make-term
(order (first-term term-list))
(negate (coeff (first-term term-list))))
(negate-terms (rest-terms term-list)))))
(define (sub-poly p1 p2)
(if (same-variable? (variable p1) (variable p2))
(make-poly
(variable p1)
(add-terms (term-list p1) (negate-terms (term-list p2))))
(error "Polys not in same var: SUB-POLY" (list p1 p2))))
;; Division
(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
L1
(negate-terms
(mul-term-by-all-terms
(make-term new-o new-c)
L2)))
L2)))
(list
(adjoin-term
(make-term new-o new-c)
(car rest-of-result))
(cadr rest-of-result))))))))
(define (div-poly p1 p2)
(if (same-variable? (variable p1) (variable p2))
(let ((ans (div-terms (term-list p1)
(term-list p2))))
(list (make-poly (variable p1)
(car ans))
(make-poly (variable p1)
(cadr ans))))
(error "Polys not in same var: DIV-POLY" (list p1 p2))))
;; interface to rest of the system
(define (tag p) (attach-tag 'polynomial p))
(put '=zero? '(polynomial) (lambda (p) (poly-zero? (term-list 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 'div '(polynomial polynomial)
(lambda (p1 p2) (tag (div-poly p1 p2))))
(put 'make 'polynomial
(lambda (var terms) (tag (make-poly var terms))))
(put 'negate '(polynomial)
(lambda (p)
(make-polynomial
(variable p)
(negate-terms (term-list p)))))
(put 'sub '(polynomial polynomial)
(lambda (p1 p2) (tag (sub-poly p1 p2))))
'done)
;; representation of terms
(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-polynomial var terms)
((get 'make 'polynomial) var terms))
; Testing
(load "arithmetic-package.scm")
(install-polynomial-package)
(define p1
(make-polynomial
'x
(list (make-term 5 (make-scheme-number 1))
(make-term 0 (make-scheme-number -1))))) ; x^5 - 1
(define p2
(make-polynomial
'x
(list (make-term 2 (make-scheme-number 1))
(make-term 0 (make-scheme-number -1))))) ; x^2 - 1
(div p1 p2)
; Note: This returns an error, couldn't get this working :/