;;; -*- Mode:Common-Lisp; Package:mma; Base:10 -*-
(in-package :mma)
;;
;; Written by: Tak W. Yan, Richard J. Fateman
;; File: rat.lisp
;; Contains: rational function (in partially-factored form) arithmetic
;;(provide 'rat1)
;;; (c) Copyright 1990, 1991 Richard J. Fateman, Tak W. Yan
#| A fpe is a (possibly) factored polynomial expression implemented
as a list of pairs of polynomials and exponents. For example,
2 2 4
12 (x + 1) (y + y - 5)
is represented as a list of three pairs, the first pair being the
the number 12 and the exponent 1, the second being the
polynomial (x + 1) and the exponent 2, and the third being the
polynomial (y^2 + y - 5) and the exponent 4.
Currently, the first pair is always an integer with exponent 1.
Although the polynomials could have coefficients in any ring
for some operations, the idea of gcd as used here assumes coefs form a
unique factorization domain, and therefore it is principally useful
to have integer coefficients. (Rationals are not a UFD since
1/2*2 = 1/3*3 = 1 etc.)
Getting back to our representation --
There is a first pair that always looks like (integer . 1).
Subsequent pairs consist of
distinct polynomials with integer coefficients, and
exponents which are integers >= 1.
It is not required that polys be squarefree, monic, irreducible
or restricted in some other ways. Examples of possible pairs:
( (x^2) . 2) ; ( (3 x^2 - 3 x) . 1)
actually, the polynomials will be in some better form, so it is
more accurate to depict these examples as
( #(1 0 0 1) . 2) ; ( #(1 0 -3 3) . 1) where x <--> 1
The polynomials in an fpe, except for intermediate forms produced
during gcd computation, are maintained in lexicographically
increasing order. Note that two polynomials in fpe form
may not be IDENTICAL even though they are mathematically
equivalent, because of different factorings.
For example, ((1 . 1) ( (x^2+2x+1) . 1)) and ((1 . 1) ((x+1) . 2))
are equivalent but not identical. If they were added together,
the result would be (if we refrained from computing
gcds)
((1 . 1) ((2x^2+4*x+2) . 1))
Another equivalent answer (more effort to compute .. we don't do
this...) would get
((2 . 1) ((x+1) . 2))
On the other hand, if we were adding ((1 . 1) ((x+1) . 2)) and
((1 . 1) ((x+1) . 3)), the common factor of x+1 would be obvious
and we would get ((1 . 1) ((x+1) . 2) ((x+2) . 1))
|#
;;(declaim (optimize (speed 3) (safety 0) (space 0) (compilation-speed 0)))
;;(declaim (optimize (speed 0) (safety 3) (space 0) (compilation-speed 0)))
(eval-when (:compile-toplevel) (load "c:/lisp/mma4max/poly.lisp")) ;; need macro defs 3/9/2019
;; export what other modules may need
#+ignore (export '(make-fpe fpe-expand fpe-coef-p fpe-coefone-p fpe-negativep
fpe-insert fpe-coefzero-p
make-rat rat-p rat make-good-rat rat-numerator
rat-denominator rat+ rat* rat^ rat/ poly2rat
*rat-gcd-level*))
;; default is to compute the gcd, but only between factors of
;; the numerator with factors of the denominator.
(defvar *rat-gcd-level* 'gcd)
(defvar *expand* nil)
;; other possible settings:
;; cdi: just find the obvious common factors by identity. That is
;; u and v have a common divisor iff u = v.
;; cdd: common divisor by division. That is, u and v have a
;; common divisor if one divides the other exactly.
;; current implementation uses gcd as default.
;; NOT IMPLEMENTED techniques:
;; supergcd: all gcds of factors in numerator OR denominator are
;; considered for pair-wise gcds. That is, (x^2-1)*(x+1) --> (x+1)^2*(x-1)
;; square-free: all factors are run through a square-free factorization
;; program. (they are tagged as square-free to save redundant calculations).
;; factors: all are factored over the integers. (also tagged)
;; make-fpe: make a "normal" fpe p^e from a polynomial p, exponent e>=0
;; examples:
;; (make-fpe poly 4) --> (( 1 . 1) (poly 4)) ;;usual case
;; (make-fpe 0 1) --> (( 0 . 1))
;; by virtue of using fpe-insert with monomial flag = t, if
;; make-fpe is given something like p = 3x^2y^3, e=2, it will
;; produce ((9 . 1) ( #(x 0 1) . 4) (#(y 0 1) . 6)).
;; {actually, you won't see x and y, but some number-encoding of them.}
(defun make-fpe (p e)
(cond ((coefp p)
(list (cons (coef^ p e) 1)))
(t (fpe-insert p e '((1 . 1)) t))))
(defun poly2rat (p e)
(cond ((>= e 0)(make-rat :numerator (make-fpe p e)
:denominator (list '(1 . 1))))
(t (make-good-rat (list '(1 . 1))
(make-fpe p (- e))))))
;; fpe-copy: return a copy of the fpe u
(defun fpe-copy (u) (copy-tree u))
;; fpe-expand: expand the fpe into a fully-expanded form; the result
;; is returned as a polynomial
(defun fpe-expand (u)
(let ((*expand* t))
(do ((ul u (cdr ul)) (p 1))
((null ul) p)
(setq p (p* p (p^ (caar ul) (cdar ul)))))))
;;; A fpe is "normal" if its first factor is an integer and it has
;;; no other integer factors. Also, the coefficient of all
;;; monomial factors should be 1. (reason: they look odd otherwise...
;;; consider 3*x^2*4*y^2 instead of 12*x^2*y^2)
;;; The factors are unique, and
;;; sorted. All coefficients in the domain
;;; are represented as c^1, including coefzero and coefone.
;; fpe-norm: normalize a fpe
;;; this program could be reprogrammed to do much less consing.
(defun fpe-norm (u &aux ctest (const 1))
(labels
((fn1(ul) ; aux function to sort terms.
(cond ((null ul) nil)
((coefp (caar ul)) ; stick coefs up front
(setq const (coef* const
(coef^ (caar ul)(cdar ul))))
(fn1 (cdr ul)))
(t
(fn1-insert (car ul) (fn1 (cdr ul))))))
(fn1-insert (h ul)
(cond ((null ul)
;; Since factor (car h) is nowhere to be seen further in
;; this fpe, make the pair of this factor and its power
;; appear at the end of the list.
(cons h ul))
((eq 'e (setq ctest (pcompare (car h) (caar ul))))
;; We found this factor. Increment the exponent
(cons (cons (car h)(+ (cdr h)(cdar ul))) (cdr ul)))
;; Otherwise we keep looking for the factor.
((eq 'l ctest)
;; this poly is less in ordering that (caar ul),
;; so place it right here.
(cons h ul))
;; This keep searching
(t
(cons (car ul)(fn1-insert h (cdr ul)))
ul)))
) ; end of local fns
(setq u (fn1 u))
(cons (cons const 1) u)))
(defun fpe-coef-p (u) (null(cdr u)))
(defun fpe-coefone-p (u) (and (null (cdr u)) (coefonep (caar u))))
(defun fpe-coefzero-p (u) (coefzerop (caar u)))
;; fpe-insert: insert a pair of polynomial and exponent into the fpe u.
;; This checks to see whether the polynomial already exists in
;; the fpe; if yes, increment the exponent; otherwise
;; adds the pair to the fpe. This will not change u.
;; in effect, this multiplies u in fpe form, by poly^u.
;; compare to fpe-*, which multiplies 2 polys in fpe form.
;; if monom is t, then maybe poly is a monomial. Check it
;; and insert it in pieces if appropriate.
(defun fpe-insert (poly exp u &optional (monom nil) &aux ctest)
(labels
((fi1(ul)
;; this routine recurses down the list of factors
(cond ((null ul)
;; Since this factor is nowhere to be seen in
;; this fpe, insert the new factor and its power
;; at the end of the list.
(list (cons poly exp)))
((eq 'e (setq ctest (pcompare poly (caar ul))))
;; We found this factor. Increment the exponent.
(cons (cons (caar ul) (+ exp (cdar ul)))(cdr ul)))
;; Otherwise we keep looking for the factor.
((eq 'l ctest)
;; this poly is less in ordering than (caar ul),
;; so place it right here.
(cons (cons poly exp) ul))
;; This keep searching
(t (cons (car ul)(fi1 (cdr ul)))))))
(cond ((coefp poly)
;; inserting an integer factor: modify the first element
(cond ((coefonep poly) u)
((coefzerop poly) (make-fpe 0 1))
(t (cons (cons (coef* (caar u)
(coef^ poly exp))
1)
(cdr u)))))
((= exp 0) u) ;;multiplying u by z^0 gives u
((and monom (monomialp poly)(or (> (degree poly) 1)
(not (coefonep (lc poly)))))
;; the poly is something like (3*x^2) ^4
;; insert 81 (i.e. 3^4) into 1*(x)^8.
;; should work recursively for (3*y^2*x^2)^4
(fpe-insert (lc poly) exp (fpe-insert
(vector (svref poly 0) 0 1)
(* (degree poly) exp) u )
monom))
;; we can, more generally, not insist on monomial,
;; but only 0 x^1 term. That is,
;; (a*x^5+b*x^2) --> x^2*(a*x^3+b). or
;; (x^5+x^2)^3 --> x^6*(x^3+1)^3. etc.
;; note that 0 constant term is not useful because
;; x is encoded as (vector n 0 1) and has 0 const term
((and monom
(not *expand*)
(coefzerop(constc poly))
(> (length (the simple-vector poly)) 3)
(coefzerop (svref poly 2)))
(do ((i 2 (1+ i)))
((null (coefzerop (svref poly i)));there's a non-0
(fpe-insert
(vector (svref poly 0) 0 1)
(* exp (1- i)) ;; the degree of the factor
(fpe-insert (polyshift poly (1- i)) exp u)))
(declare (fixnum i))
;; no do-body
))
(t (fi1 u)))))
(defun polyshift(p i) ;; this divides a poly in x by x^i
(let ((z(subseq p i)))
(setf (svref z 0) (svref p 0)) z))
;; fpe-* multiplies two fpe's u and v. To be canonical,
;;identical polynomials will appear only once in the result.
;; Both u and v are preserved. No wasted conses.
(defun fpe-* (u v &aux ctest)
(labels ((fm1(ul vl)
(cond ((null ul) vl)
((null vl) ul)
((eq 'e (setq ctest (pcompare (caar ul) (caar vl))))
(cons (cons (caar ul) (+ (cdar ul)(cdar vl)))
(fm1 (cdr ul)(cdr vl))))
((eq 'l ctest)
;; (caar ul) is less in ordering that (caar vl),
;; so place it right here.
(cons (car ul) (fm1 (cdr ul) vl)))
;; Otherwise (car vl) goes here
(t (cons (car vl)(fm1 ul (cdr vl)))))))
(cond ((and (null (cdr u)) (coefonep (caar u))) v) ;; u is coefone
((and (null (cdr v)) (coefonep (caar v))) u) ;; v is coefone
(t (cons (cons (coef* (caar u)(caar v)) 1) ; mult consts.
(fm1 (cdr u)(cdr v)))))))
(defun fpe-^ (a n) ;;powering an fpe is "easy"
;; presumably n is a positive integer, although if it is some
;; other number, this program won't break.
(labels((fp^1 (a)
(cond ((null a)nil)
(t (cons (cons (caar a) (* n (cdar a)))
(fp^1 (cdr a)))))))
(cons (cons (expt (caar a) n) 1)
(fp^1 (cdr a)))))
(defun fpe-negative-p (u) (coef-negative-p (caar u)))
(defun fpe-negate (u)
(cons (cons (coefneg (caar u))(cdar u))(cdr u)))
;;; These procedures are concerned with finding the gcd and cd of fpe's.
;;; They are independent of the implementation of fpe's.
;; fpe-gcd-cofac:
;; returns 3 values: g= the gcd of u and v;
;; ubyg = u/g;
;; vbyg = v/g.
;; u and v are unchanged, though any of the outputs may share structure.
;; all values are fpe in normal form.
(defun fpe-gcd-cofac (u v)
(declare(optimize (safety 3)))
(setq u (append u nil) v (append v nil));; copy only top levels of inputs
(let((g '((1 . 1)))) ; g= gcd = 1, initially
;; (format t "~% in fpe-gcd-cofac: uvg ~s ~s ~s" u v g)
(do ((i u (cdr i)))
;; for each polynomial in u do the following
;; until we run off the end of u. Then return the triple
;; .
((null i) (values g (fpe-norm u)(fpe-norm v)))
;; (format t "~% do in fpe-gcd-cofac: uvgi ~s ~s ~s ~s" u v g i)
(if (not(coefonep (caar i))) ;; do this only if (caar i) is not 1
(do ((j v (cdr j)));; for each polynomial in v do the following
((null j))
;; (format t "~% j=~s"j)
(if (not (coefonep (caar j))) ;otherwise next j
(let* ((up (caar i)) ; next poly in u
(vp (caar j)) ; next poly in v
(ue (cdar i))
(ve (cdar j)))
;; (format t "~% up vp ue ve ~s ~s ~s ~s" up vp ue ve)
(multiple-value-bind
(gp uq vq) ;gcd, up/gp, vp/gp
(pgcdswitch-cofac up vp)
;; (format t "~% gp uq vq ~s ~s ~s " gp uq vq)
(cond ((coefonep gp)) ; no non-trivial divisor! just advance i,j.
(t
;; ok, we've found a common factor gp
(let ((ge (min ue ve)));; gcd is really gp^ge
;; add the factor to the ones already found.
;; incidentally, this factor may already be
;; in g because of some earlier discovered gcd.
(setq g (fpe-insert gp ge g t));; g = g*gp^ge
;; blot out the two factors in u and v.
;; we may have to put some remnants at the end, though.
;; splice in what's left of powers of up and powers
;; of vp, if any, and then consider up/gp and vp/gp
;; possible missed factorization here.. if we compute
;; gcd of x^2-1 and (x^2*y^2-y^2)^3 we get a
;; co-factor of y^2. This is a monomial. It would be
;; better to put y^6 on the factor list, rather than
;; (y^2)^3. Could check in fpe-norm for monomials,
;; but this would add to expense.
(cond ((= ge ue)
(setf (car i) (cons uq ue))
(setq up uq))
(t ; that is, ge < ue
;; first decrement the exponent on this factor
;; to make it reflect the number of times
;; we've divided out gp
(setf (car i)(cons gp (- ue ge)))
(setq up gp)
(if (coefonep uq) nil
(nconc i (list (cons uq ue))))))
(cond ((= ge ve)
(setf (car j) (cons vq ve))
(setq vp vq))
(t ; that is, ge < ve
(setf (car j)(cons gp (- ve ge)))
(setq vp gp)
(if (coefonep vq) nil
(nconc j (list (cons vq ve))))))
)))))))))))
(defun pgcdswitch-cofac(u v)
(case *rat-gcd-level*
(gcd
;;compute the gcd. Which algorithm depends perhaps
;; on other switches in polynomial package.
(pgcd-cofac u v))
(cdi
;; look for identical common factors only
;; equalp checks element by element in arrays
(if (equalp u v)(values u 1 1)(values 1 u v)))
(cdd
;; look to see if u divides v or vice versa
(let (r)
(cond ((setq r (p/-test u v)) (values v r 1))
((setq r (p/-test v u)) (values u 1 r))
(t (values 1 u v)))))
;; put other choices here
(t ; use gcd as default
(pgcd-cofac u v))))
;;; A rat is a rational polynomial implemented as a structure
;;; consisting of two fpe's: the numerator and the denominator.
(defstruct rat numerator denominator)
;;; A "good" rat is one whose denominator is always positive.
;;; If the rat as a whole is negative, the negative sign
;;; is in the numerator.
;; make-good-rat
(defun make-good-rat (n d)
(cond ((fpe-negative-p d)
(setq d (fpe-negate d))
(setq n (fpe-negate n))))
(make-rat :numerator n :denominator d))
;;; The following procedures perform simple arithmetic on rats.
;;; The returned rats from these procedures will be "normal,"
;;; provided that the arguments are "normal." A rat is said to
;;; be "normal" if
;;; 1) it is in reduced form (no common factors between numerator
;;; and denominator).
;;; 2) each polynomial in a fpe that makes up the numerator
;;; or the denominator appears only once within that rat.
;;; 3) the fpe's are themselves normal: leading coefficient followed
;;; by terms sorted etc.
;; rat*: non-destructively multiply two rats r1 and r2
;; reference - W. S. Brown's paper
(defun rat* (r1 r2) ;; r1 * r2 = a/b * c/d
(let ((a (rat-numerator r1))
(b (rat-denominator r1))
(c (rat-numerator r2))
(d (rat-denominator r2))
g num den)
;; we assume that gcd(a,b)=1, and also that gcd(c,d)=1.
;; This may not be, strictly speaking, true, depending on *rat-gcd-level*.
(if (or (fpe-coefzero-p a)
(fpe-coefzero-p c))
(return-from rat* (poly2rat 0 1)))
;; First, if a and c have any factors in common,
;; set them aside, because if g is a factor of a, then gcd(g,b)
;; is predictably 1, so no need to compute it.
;; We could set aside all common factors regardless of multiplicity
;; as an additional optimization. (Tak -- this is what you were
;; doing, I guess). An alternative would be to save all poly gcd
;; results in an eq-hash table..what do you think? We'd have
;; to be careful not to make equal but not-eq polynomials... RJF
(multiple-value-setq (num a c) (fpe-gcd-cofac a c))
;; Ditto for b and d
(multiple-value-setq (den b d) (fpe-gcd-cofac b d))
;; we can conclude that gcd(num,den)=1 by construction, and
;; that with these new values, r1*r2 = (num^2*a*c)/(den^2*b*d),
;; subject, however, to more gcd removal. Next,
;; remove the common factors from a and d, then from b and c.
(multiple-value-setq (g a d)(fpe-gcd-cofac a d))
(multiple-value-setq (g b c)(fpe-gcd-cofac b c))
;; put back the factors that were removed
(setq a (fpe-* a (fpe-^ num 2)))
(setq b (fpe-* b (fpe-^ den 2)))
;; finally put the pieces together
(make-good-rat (fpe-* a c) (fpe-* b d))))
;; division. almost same as *.
(defun rat/ (r1 r2) ;; r1 * r2 = a/b / d/c
(let ((a (rat-numerator r1))
(b (rat-denominator r1))
(d (rat-numerator r2)) ;; note change from rat*
(c (rat-denominator r2)) ;; ditto
g num den)
(if (fpe-coefzero-p a) (return-from rat/ (poly2rat 0 1)))
(if (fpe-coefzero-p d) (error "Rational division by zero"))
(multiple-value-setq (num a c) (fpe-gcd-cofac a c))
(multiple-value-setq (den b d) (fpe-gcd-cofac b d))
(multiple-value-setq (g a d)(fpe-gcd-cofac a d))
(multiple-value-setq (g b c)(fpe-gcd-cofac b c))
(setq a (fpe-* a (fpe-^ num 2)))
(setq b (fpe-* b (fpe-^ den 2)))
(make-good-rat (fpe-* a c) (fpe-* b d))))
;; rat+: non-destructively add two rats r1 and r2
(defun rat+ (r1 r2)
(let ((a (rat-numerator r1))
(b (rat-denominator r1))
(c (rat-numerator r2)) ;; want to perform
(d (rat-denominator r2)) ;; a/b + c/d
(n (make-fpe (coefone) 1))
g num den)
(if (fpe-coefzero-p a) (return-from rat+ r2))
(if (fpe-coefzero-p c) (return-from rat+ r1))
;; extract common factors from a and b, also b and d.
(multiple-value-setq
(num a c)(fpe-gcd-cofac a c))
(multiple-value-setq
(den b d)(fpe-gcd-cofac b d))
;; now r1+r2 = (num/den)* (a/b+c/d) and gcd(b,d)=1.
;; n=ad+bc
;; (format t "~%check0 ~s~%"n)
(setq n (pnorm(p+ (fpe-expand (fpe-* a d)); n is a polynomial
(fpe-expand (fpe-* b c)))))
;; (format t "~%check ~s~%"n)
(if (coefzerop n) (return-from rat+ (make-rat :numerator '((0 . 1))
:denominator '((1 . 1)))))
(setq n (make-fpe n 1)) ; now n is an fpe.
(setq den(fpe-* den (fpe-* b d))) ;; set denom to bd * den
;; remove common factors between ad+bc and bd*den
(multiple-value-setq (g n den)(fpe-gcd-cofac n den))
(setq num (fpe-* num n)); set num to num *(ad + bc)
(make-good-rat num den)))
;; rat^: raise r to the power e
(defun rat^ (r e)
(cond ((= e 0) (make-rat :numerator '((1 . 1))
:denominator '((1 . 1))))
((> e 0)
(make-rat :numerator (fpe-^ (rat-numerator r) e)
:denominator (fpe-^ (rat-denominator r) e)))
(t (if (fpe-coefzero-p (rat-numerator r))
(error "Rational division by zero"))
(make-good-rat ;check sign of denom..
(fpe-^ (rat-denominator r) (- e))
(fpe-^ (rat-numerator r) (- e))))))