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1;;----------------------------------------------------------------
2;;; -*- Mode: Lisp -*-
3;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
4;;;
5;;; Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>
6;;;
7;;; This program is free software; you can redistribute it and/or modify
8;;; it under the terms of the GNU General Public License as published by
9;;; the Free Software Foundation; either version 2 of the License, or
10;;; (at your option) any later version.
11;;;
12;;; This program is distributed in the hope that it will be useful,
13;;; but WITHOUT ANY WARRANTY; without even the implied warranty of
14;;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15;;; GNU General Public License for more details.
16;;;
17;;; You should have received a copy of the GNU General Public License
18;;; along with this program; if not, write to the Free Software
19;;; Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
20;;;
21;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
22
23(defpackage "POLYNOMIAL"
24 (:use :cl :utils :monom :copy)
25 (:export "POLY"
26 "POLY-DIMENSION"
27 "POLY-TERMLIST"
28 "POLY-TERM-ORDER"
29 "POLY-INSERT-TERM"
30 "SCALAR-MULTIPLY-BY"
31 "SCALAR-DIVIDE-BY"
32 "LEADING-TERM"
33 "LEADING-MONOMIAL"
34 "LEADING-COEFFICIENT"
35 "SECOND-LEADING-TERM"
36 "SECOND-LEADING-MONOMIAL"
37 "SECOND-LEADING-COEFFICIENT"
38 "ADD-TO"
39 "ADD"
40 "SUBTRACT-FROM"
41 "SUBTRACT"
42 "CHANGE-TERM-ORDER"
43 "STANDARD-EXTENSION"
44 "STANDARD-EXTENSION-1"
45 "STANDARD-SUM"
46 "SATURATION-EXTENSION"
47 "ALIST->POLY"
48 "->INFIX"
49 "UNIVERSAL-EZGCD"
50 "S-POLYNOMIAL"
51 "POLY-CONTENT"
52 "POLY-PRIMITIVE-PART"
53 "SATURATION-EXTENSION-1"
54 "MAKE-POLY-VARIABLE"
55 "MAKE-POLY-CONSTANT"
56 "UNIVERSAL-EXPT"
57 "UNIVERSAL-EQUALP"
58 "POLY-LENGTH"
59 "POLY-P"
60 "+LIST-MARKER+"
61 "POLY-EVAL")
62 (:documentation "Implements polynomials. A polynomial is essentially
63a mapping of monomials of the same degree to coefficients. The
64momomials are ordered according to a monomial order."))
65
66(in-package :polynomial)
67
68(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
69
70(defclass poly ()
71 ((dimension :initform nil
72 :initarg :dimension
73 :accessor poly-dimension
74 :documentation "Shared dimension of all terms, the number of variables")
75 (termlist :initform nil :initarg :termlist :accessor poly-termlist
76 :documentation "List of terms.")
77 (order :initform #'lex> :initarg :order :accessor poly-term-order
78 :documentation "Monomial/term order."))
79 (:default-initargs :dimension nil :termlist nil :order #'lex>)
80 (:documentation "A polynomial with a list of terms TERMLIST, ordered
81according to term order ORDER, which defaults to LEX>."))
82
83(defmethod print-object ((self poly) stream)
84 (print-unreadable-object (self stream :type t :identity t)
85 (with-accessors ((dimension poly-dimension)
86 (termlist poly-termlist)
87 (order poly-term-order))
88 self
89 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
90 dimension termlist order))))
91
92(defgeneric change-term-order (self other)
93 (:documentation "Change term order of SELF to the term order of OTHER.")
94 (:method ((self poly) (other poly))
95 (unless (eq (poly-term-order self) (poly-term-order other))
96 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
97 (poly-term-order self) (poly-term-order other)))
98 self))
99
100(defgeneric poly-insert-term (self term)
101 (:documentation "Insert a term TERM into SELF before all other
102 terms. Order is not enforced.")
103 (:method ((self poly) (term term))
104 (cond ((null (poly-dimension self))
105 (setf (poly-dimension self) (monom-dimension term)))
106 (t (assert (= (poly-dimension self) (monom-dimension term)))))
107 (push term (poly-termlist self))
108 self))
109
110(defgeneric poly-append-term (self term)
111 (:documentation "Append a term TERM to SELF after all other terms. Order is not enforced.")
112 (:method ((self poly) (term term))
113 (cond ((null (poly-dimension self))
114 (setf (poly-dimension self) (monom-dimension term)))
115 (t (assert (= (poly-dimension self) (monom-dimension term)))))
116 (setf (cdr (last (poly-termlist self))) (list term))
117 self))
118
119(defun alist->poly (alist &aux (poly (make-instance 'poly)))
120 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
121It can be used to enter simple polynomials by hand, e.g the polynomial
122in two variables, X and Y, given in standard notation as:
123
124 3*X^2*Y^3+2*Y+7
125
126can be entered as
127(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
128
129NOTE: The primary use is for low-level debugging of the package."
130 (dolist (x alist poly)
131 (poly-insert-term poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))
132
133(defmethod update-instance-for-different-class :after ((old term) (new poly) &key)
134 "Converts OLD of class TERM to a NEW of class POLY, by making it into a 1-element TERMLIST."
135 (reinitialize-instance new
136 :dimension (monom-dimension old)
137 :termlist (list old)))
138
139(defmethod update-instance-for-different-class :after ((old monom) (new poly) &key)
140 "Converts OLD of class MONOM to a NEW of class POLY, by making it into a 1-element TERMLIST."
141 (reinitialize-instance new
142 :dimension (monom-dimension old)
143 :termlist (list (change-class old 'term))))
144
145(defmethod universal-equalp ((self poly) (other poly))
146 "Implements equality of polynomials."
147 (and (eql (poly-dimension self) (poly-dimension other))
148 (every #'universal-equalp (poly-termlist self) (poly-termlist other))
149 (eq (poly-term-order self) (poly-term-order other))))
150
151(defgeneric leading-term (object)
152 (:method ((self poly))
153 (car (poly-termlist self)))
154 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
155
156(defgeneric second-leading-term (object)
157 (:method ((self poly))
158 (cadar (poly-termlist self)))
159 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
160
161(defgeneric leading-monomial (object)
162 (:method ((self poly))
163 (change-class (copy-instance (leading-term self)) 'monom))
164 (:documentation "The leading monomial of a polynomial, or NIL for zero polynomial."))
165
166(defgeneric second-leading-monomial (object)
167 (:method ((self poly))
168 (change-class (copy-instance (second-leading-term self)) 'monom))
169 (:documentation "The leading monomial of a polynomial, or NIL for zero polynomial."))
170
171(defgeneric leading-coefficient (object)
172 (:method ((self poly))
173 (term-coeff (leading-term self)))
174 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
175
176(defgeneric second-leading-coefficient (object)
177 (:method ((self poly))
178 (term-coeff (second-leading-term self)))
179 (:documentation "The second leading coefficient of a polynomial. It
180 signals error for a polynomial with at most one term."))
181
182(defmethod universal-zerop ((self poly))
183 "Return T iff SELF is a zero polynomial."
184 (null (poly-termlist self)))
185
186(defgeneric poly-length (self)
187 (:documentation "Return the number of terms.")
188 (:method ((self poly))
189 (length (poly-termlist self))))
190
191(defgeneric scalar-multiply-by (self other)
192 (:documentation "Multiply vector SELF by a scalar OTHER.")
193 (:method ((self poly) other)
194 (mapc #'(lambda (term) (setf (term-coeff term) (multiply (term-coeff term) other)))
195 (poly-termlist self))
196 self))
197
198(defgeneric scalar-divide-by (self other)
199 (:documentation "Divide vector SELF by a scalar OTHER.")
200 (:method ((self poly) other)
201 (mapc #'(lambda (term) (setf (term-coeff term) (divide (term-coeff term) other)))
202 (poly-termlist self))
203 self))
204
205(defmethod multiply-by ((self poly) (other monom))
206 "Multiply a polynomial SELF by OTHER."
207 (mapc #'(lambda (term) (multiply-by term other))
208 (poly-termlist self))
209 self)
210
211(defmethod multiply-by ((self poly) (other term))
212 "Multiply a polynomial SELF by OTHER."
213 (mapc #'(lambda (term) (multiply-by term other))
214 (poly-termlist self))
215 self)
216
217(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
218 "Return an expression which will efficiently adds/subtracts two
219polynomials, P and Q. The addition/subtraction of coefficients is
220performed by calling ADD/SUBTRACT-FN. If UMINUS-FN is supplied, it is
221used to negate the coefficients of Q which do not have a corresponding
222coefficient in P. The code implements an efficient algorithm to add
223two polynomials represented as sorted lists of terms. The code
224destroys both arguments, reusing the terms to build the result."
225 `(macrolet ((lc (x) `(term-coeff (car ,x))))
226 (do ((p ,p)
227 (q ,q)
228 r)
229 ((or (endp p) (endp q))
230 ;; NOTE: R contains the result in reverse order. Can it
231 ;; be more efficient to produce the terms in correct order?
232 (unless (endp q)
233 ;; Upon subtraction, we must change the sign of
234 ;; all coefficients in q
235 ,@(when uminus-fn
236 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
237 (setf r (nreconc r q)))
238 (unless (endp p)
239 (setf r (nreconc r p)))
240 r)
241 (multiple-value-bind
242 (greater-p equal-p)
243 (funcall ,order-fn (car p) (car q))
244 (cond
245 (greater-p
246 (rotatef (cdr p) r p)
247 )
248 (equal-p
249 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
250 (cond
251 ((universal-zerop s)
252 (setf p (cdr p))
253 )
254 (t
255 (setf (lc p) s)
256 (rotatef (cdr p) r p))))
257 (setf q (cdr q))
258 )
259 (t
260 ;;Negate the term of Q if UMINUS provided, signallig
261 ;;that we are doing subtraction
262 ,(when uminus-fn
263 `(setf (lc q) (funcall ,uminus-fn (lc q))))
264 (rotatef (cdr q) r q))))
265 ;;(format t "P:~A~%" p)
266 ;;(format t "Q:~A~%" q)
267 ;;(format t "R:~A~%" r)
268 )))
269
270
271
272(defgeneric add-to (self other)
273 (:documentation "Add OTHER to SELF.")
274 (:method ((self number) (other number))
275 (+ self other))
276 (:method ((self poly) (other number))
277 (add-to self (make-poly-constant (poly-dimension self) other)))
278 (:method ((self number) (other poly))
279 (add-to (make-poly-constant (poly-dimension other) self) other)))
280
281
282(defgeneric subtract-from (self other)
283 (:documentation "Subtract OTHER from SELF.")
284 (:method ((self number) (other number))
285 (- self other))
286 (:method ((self poly) (other number))
287 (subtract-from self (make-poly-constant (poly-dimension self) other))))
288
289
290#|
291(defmacro def-add/subtract-method (add/subtract-method-name
292 uminus-method-name
293 &optional
294 (doc-string nil doc-string-supplied-p))
295 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
296 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
297 ,@(when doc-string-supplied-p `(,doc-string))
298 ;; Ensure orders are compatible
299 (change-term-order other self)
300 (setf (poly-termlist self) (fast-add/subtract
301 (poly-termlist self) (poly-termlist other)
302 (poly-term-order self)
303 #',add/subtract-method-name
304 ,(when uminus-method-name `(function ,uminus-method-name))))
305 self))
306
307(eval-when (:load-toplevel :execute)
308
309 (def-add/subtract-method add-to nil
310 "Adds to polynomial SELF another polynomial OTHER.
311This operation destructively modifies both polynomials.
312The result is stored in SELF. This implementation does
313no consing, entirely reusing the sells of SELF and OTHER.")
314
315 (def-add/subtract-method subtract-from unary-minus
316 "Subtracts from polynomial SELF another polynomial OTHER.
317This operation destructively modifies both polynomials.
318The result is stored in SELF. This implementation does
319no consing, entirely reusing the sells of SELF and OTHER.")
320 )
321
322|#
323
324(defmethod unary-minus ((self poly))
325 "Destructively modifies the coefficients of the polynomial SELF,
326by changing their sign."
327 (mapc #'unary-minus (poly-termlist self))
328 self)
329
330(defun add-termlists (p q order-fn)
331 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
332 (fast-add/subtract p q order-fn #'add-to nil))
333
334(defun subtract-termlists (p q order-fn)
335 "Destructively subtracts two termlists P and Q ordered according to ORDER-FN."
336 (fast-add/subtract p q order-fn #'subtract-from #'unary-minus))
337
338(defmethod add-to ((self poly) (other poly))
339 "Adds to polynomial SELF another polynomial OTHER.
340This operation destructively modifies both polynomials.
341The result is stored in SELF. This implementation does
342no consing, entirely reusing the sells of SELF and OTHER."
343 (change-term-order other self)
344 (setf (poly-termlist self) (add-termlists
345 (poly-termlist self) (poly-termlist other)
346 (poly-term-order self)))
347 self)
348
349
350(defmethod subtract-from ((self poly) (other poly))
351 "Subtracts from polynomial SELF another polynomial OTHER.
352This operation destructively modifies both polynomials.
353The result is stored in SELF. This implementation does
354no consing, entirely reusing the sells of SELF and OTHER."
355 (change-term-order other self)
356 (setf (poly-termlist self) (subtract-termlists
357 (poly-termlist self) (poly-termlist other)
358 (poly-term-order self)))
359 self)
360
361(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
362 &optional (reverse-arg-order-P nil))
363 "Multiplies term TERM by a list of term, TERMLIST.
364Takes into accound divisors of zero in the ring, by
365deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
366is T, change the order of arguments; this may be important
367if we extend the package to non-commutative rings."
368 `(mapcan #'(lambda (other-term)
369 (let ((prod (multiply
370 ,@(cond
371 (reverse-arg-order-p
372 `(other-term ,term))
373 (t
374 `(,term other-term))))))
375 (cond
376 ((universal-zerop prod) nil)
377 (t (list prod)))))
378 ,termlist))
379
380(defun multiply-termlists (p q order-fn)
381 "A version of polynomial multiplication, operating
382directly on termlists."
383 (cond
384 ((or (endp p) (endp q))
385 ;;p or q is 0 (represented by NIL)
386 nil)
387 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
388 ((endp (cdr p))
389 (multiply-term-by-termlist-dropping-zeros (car p) q))
390 ((endp (cdr q))
391 (multiply-term-by-termlist-dropping-zeros (car q) p t))
392 (t
393 (cons (multiply (car p) (car q))
394 (add-termlists
395 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
396 (multiply-termlists (cdr p) q order-fn)
397 order-fn)))))
398
399(defmethod multiply-by ((self poly) (other poly))
400 (change-term-order other self)
401 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
402 (poly-termlist other)
403 (poly-term-order self)))
404 self)
405
406(defgeneric add-2 (object1 object2)
407 (:documentation "Non-destructively add OBJECT1 to OBJECT2.")
408 (:method ((object1 t) (object2 t))
409 (add-to (copy-instance object1) (copy-instance object2))))
410
411(defun add (&rest summands)
412 "Non-destructively adds list SUMMANDS."
413 (cond ((endp summands) 0)
414 (t (reduce #'add-2 summands))))
415
416(defun subtract (minuend &rest subtrahends)
417 "Non-destructively subtract MINUEND and SUBTRAHENDS."
418 (cond ((endp subtrahends) (unary-minus minuend))
419 (t (subtract-from (copy-instance minuend) (reduce #'add subtrahends)))))
420
421(defmethod left-tensor-product-by ((self poly) (other monom))
422 (setf (poly-termlist self)
423 (mapcan #'(lambda (term)
424 (let ((prod (left-tensor-product-by term other)))
425 (cond
426 ((universal-zerop prod) nil)
427 (t (list prod)))))
428 (poly-termlist self)))
429 (incf (poly-dimension self) (monom-dimension other))
430 self)
431
432(defmethod right-tensor-product-by ((self poly) (other monom))
433 (setf (poly-termlist self)
434 (mapcan #'(lambda (term)
435 (let ((prod (right-tensor-product-by term other)))
436 (cond
437 ((universal-zerop prod) nil)
438 (t (list prod)))))
439 (poly-termlist self)))
440 (incf (poly-dimension self) (monom-dimension other))
441 self)
442
443
444(defun standard-extension (plist &aux (k (length plist)) (i 0))
445 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
446is a list of polynomials. Destructively modifies PLIST elements."
447 (mapc #'(lambda (poly)
448 (left-tensor-product-by
449 poly
450 (prog1
451 (make-monom-variable k i)
452 (incf i))))
453 plist))
454
455(defun standard-extension-1 (plist
456 &aux
457 (plist (standard-extension plist))
458 (nvars (poly-dimension (car plist))))
459 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
460Firstly, new K variables U1, U2, ..., UK, are inserted into each
461polynomial. Subsequently, P1, P2, ..., PK are destructively modified
462tantamount to replacing PI with UI*PI-1. It assumes that all
463polynomials have the same dimension, and only the first polynomial
464is examined to determine this dimension."
465 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
466 ;; 1 from each polynomial; since UI*PI has no constant term,
467 ;; we just need to append the constant term at the end
468 ;; of each termlist.
469 (flet ((subtract-1 (p)
470 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
471 (setf plist (mapc #'subtract-1 plist)))
472 plist)
473
474
475(defun standard-sum (plist
476 &aux
477 (plist (standard-extension plist))
478 (nvars (poly-dimension (car plist))))
479 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
480Firstly, new K variables, U1, U2, ..., UK, are inserted into each
481polynomial. Subsequently, P1, P2, ..., PK are destructively modified
482tantamount to replacing PI with UI*PI, and the resulting polynomials
483are added. Finally, 1 is subtracted. It should be noted that the term
484order is not modified, which is equivalent to using a lexicographic
485order on the first K variables."
486 (flet ((subtract-1 (p)
487 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
488 (subtract-1
489 (make-instance
490 'poly
491 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
492
493(defgeneric universal-ezgcd (x y)
494 (:documentation "Solves the diophantine system: X=C*X1, Y=C*X2,
495C=GCD(X,Y). It returns C, X1 and Y1. The result may be obtained by
496the Euclidean algorithm.")
497 (:method ((x integer) (y integer)
498 &aux (c (gcd x y)))
499 (values c (/ x c) (/ y c)))
500 )
501
502(defgeneric s-polynomial (object1 object2)
503 (:documentation "Yields the S-polynomial of OBJECT1 and OBJECT2.")
504 (:method ((f poly) (g poly))
505 (let* ((lcm (universal-lcm (leading-monomial f) (leading-monomial g)))
506 (mf (divide lcm (leading-monomial f)))
507 (mg (divide lcm (leading-monomial g))))
508 (multiple-value-bind (c cf cg)
509 (universal-ezgcd (leading-coefficient f) (leading-coefficient g))
510 (declare (ignore c))
511 (subtract
512 (multiply f (change-class mf 'term :coeff cg))
513 (multiply g (change-class mg 'term :coeff cf)))))))
514
515(defgeneric poly-content (object)
516 (:documentation "Greatest common divisor of the coefficients of the polynomial object OBJECT.")
517 (:method ((self poly))
518 (reduce #'universal-gcd
519 (mapcar #'term-coeff (rest (poly-termlist self)))
520 :initial-value (leading-coefficient self))))
521
522(defun poly-primitive-part (object)
523 "Divide polynomial OBJECT by gcd of its
524coefficients. Return the resulting polynomial."
525 (scalar-divide-by object (poly-content object)))
526
527(defun poly-insert-variables (self k)
528 (left-tensor-product-by self (make-instance 'monom :dimension k)))
529
530(defun saturation-extension (f plist &aux (k (length plist)))
531 "Calculate [F', U1*P1-1,U2*P2-1,...,UK*PK-1], where
532PLIST=[P1,P2,...,PK] and F' is F with variables U1,U2,...,UK inserted
533as first K variables. It destructively modifies F and PLIST."
534 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
535 (standard-extension-1 plist)))
536
537(defun polysaturation-extension (f plist &aux (k (length plist)))
538 "Calculate [F', U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]
539and F' is F with variables U1,U2,...,UK inserted as first K
540variables. It destructively modifies F and PLIST."
541 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
542 (list (standard-sum plist))))
543
544(defun saturation-extension-1 (f p)
545 "Given family of polynomials F and a polynomial P, calculate [F',
546U*P-1], where F' is F with variable inserted as the first variable. It
547destructively modifies F and P."
548 (polysaturation-extension f (list p)))
549
550(defmethod multiply-by ((object1 number) (object2 poly))
551 (scalar-multiply-by (copy-instance object2) object1))
552
553(defun make-poly-variable (nvars pos &optional (power 1))
554 (change-class (make-monom-variable nvars pos power) 'poly))
555
556(defun make-poly-constant (nvars coeff)
557 (change-class (make-term-constant nvars coeff) 'poly))
558
559(defgeneric universal-expt (x y)
560 (:documentation "Raises X to power Y.")
561 (:method ((x number) (y integer)) (expt x y))
562 (:method ((x t) (y integer))
563 (declare (type fixnum y))
564 (cond
565 ((minusp y) (error "universal-expt: Negative exponent."))
566 ((universal-zerop x) (if (zerop y) 1))
567 (t
568 (do ((k 1 (ash k 1))
569 (q x (multiply q q)) ;keep squaring
570 (p 1 (if (not (zerop (logand k y))) (multiply p q) p)))
571 ((> k y) p)
572 (declare (fixnum k)))))))
573
574(defgeneric poly-p (object)
575 (:documentation "Checks if an object is a polynomial.")
576 (:method ((self poly)) t)
577 (:method ((self t)) nil))
578
579(defmethod ->infix :before ((self poly) &optional vars)
580 "Ensures that the number of variables in VARS maches the polynomial dimension of the
581polynomial SELF."
582 (assert (= (length vars) (poly-dimension self))))
583
584(defmethod ->infix ((self poly) &optional vars)
585 "Converts a polynomial SELF to a sexp."
586 (cons '+ (mapcar #'(lambda (x) (->infix x vars))
587 (poly-termlist self))))
588
589(defparameter +list-marker+ :[
590 "A sexp with this head is considered a list of polynomials.")
591
592(defmethod ->infix ((self cons) &optional vars)
593 (assert (eql (car self) +list-marker+))
594 (cons +list-marker+ (mapcar #'(lambda (p) (->infix p vars)) (cdr self))))
595
596
597(defun poly-eval (expr vars order)
598 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
599variables VARS. Return the resulting polynomial or list of
600polynomials. Standard arithmetical operators in form EXPR are
601replaced with their analogues in the ring of polynomials, and the
602resulting expression is evaluated, resulting in a polynomial or a list
603of polynomials in internal form. A similar operation in another computer
604algebra system could be called 'expand' or so."
605 (labels ((p-eval (p) (poly-eval p vars order))
606 (p-eval-scalar (p) (poly-eval p '() order))
607 (p-eval-list (plist) (mapcar #'p-eval plist)))
608 (cond
609 ((eq expr 0)
610 (make-instance 'poly :dimension (length vars)))
611 ((member expr vars :test #'equalp)
612 (let ((pos (position expr vars :test #'equalp)))
613 (make-poly-variable (length vars) pos)))
614 ((atom expr)
615 expr)
616 ((eq (car expr) +list-marker+)
617 (cons +list-marker+ (p-eval-list (cdr expr))))
618 (t
619 (case (car expr)
620 (+ (reduce #'add (p-eval-list (cdr expr))))
621 (- (apply #'subtract (p-eval-list (cdr expr))))
622 (*
623 (if (endp (cddr expr)) ;unary
624 (p-eval (cadr expr))
625 (reduce #'multiply (p-eval-list (cdr expr)))))
626 (/
627 ;; A polynomial can be divided by a scalar
628 (cond
629 ((endp (cddr expr))
630 ;; A special case (/ ?), the inverse
631 (divide (cadr expr)))
632 (t
633 (let ((num (p-eval (cadr expr)))
634 (denom-inverse (apply #'divide (mapcar #'p-eval-scalar (cddr expr)))))
635 (multiply denom-inverse num)))))
636 (expt
637 (cond
638 ((member (cadr expr) vars :test #'equalp)
639 ;;Special handling of (expt var pow)
640 (let ((pos (position (cadr expr) vars :test #'equalp)))
641 (make-poly-variable (length vars) pos (caddr expr))))
642 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
643 ;; Negative power means division in coefficient ring
644 ;; Non-integer power means non-polynomial coefficient
645 expr)
646 (t (universal-expt (p-eval (cadr expr)) (caddr expr)))))
647 (otherwise
648 expr))))))
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