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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 "MAKE-ZERO-FOR"
57 "MAKE-UNIT-FOR"
58 "UNIVERSAL-EXPT"
59 "UNIVERSAL-EQUALP"
60 "POLY-LENGTH"
61 "POLY-REVERSE"
62 "POLY-P"
63 "+LIST-MARKER+"
64 "POLY-EVAL")
65 (:documentation "Implements polynomials. A polynomial is essentially
66a mapping of monomials of the same degree to coefficients. The
67momomials are ordered according to a monomial order."))
68
69(in-package :polynomial)
70
71(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
72
73(defclass poly ()
74 ((dimension :initform nil
75 :initarg :dimension
76 :accessor poly-dimension
77 :documentation "Shared dimension of all terms, the number of variables")
78 (termlist :initform nil :initarg :termlist :accessor poly-termlist
79 :documentation "List of terms.")
80 (order :initform #'lex> :initarg :order :accessor poly-term-order
81 :documentation "Monomial/term order."))
82 (:default-initargs :dimension nil :termlist nil :order #'lex>)
83 (:documentation "A polynomial with a list of terms TERMLIST, ordered
84according to term order ORDER, which defaults to LEX>."))
85
86(defmethod print-object ((self poly) stream)
87 (print-unreadable-object (self stream :type t :identity t)
88 (with-accessors ((dimension poly-dimension)
89 (termlist poly-termlist)
90 (order poly-term-order))
91 self
92 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
93 dimension termlist order))))
94
95(defgeneric change-term-order (self other)
96 (:documentation "Change term order of SELF to the term order of OTHER.")
97 (:method ((self poly) (other poly))
98 (unless (eq (poly-term-order self) (poly-term-order other))
99 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
100 (poly-term-order self) (poly-term-order other)))
101 self))
102
103(defgeneric poly-insert-term (self term)
104 (:documentation "Insert a term TERM into SELF before all other
105 terms. Order is not enforced.")
106 (:method ((self poly) (term term))
107 (cond ((null (poly-dimension self))
108 (setf (poly-dimension self) (monom-dimension term)))
109 (t (assert (= (poly-dimension self) (monom-dimension term)))))
110 (push term (poly-termlist self))
111 self))
112
113(defgeneric poly-append-term (self term)
114 (:documentation "Append a term TERM to SELF after all other terms. Order is not enforced.")
115 (:method ((self poly) (term term))
116 (cond ((null (poly-dimension self))
117 (setf (poly-dimension self) (monom-dimension term)))
118 (t (assert (= (poly-dimension self) (monom-dimension term)))))
119 (setf (cdr (last (poly-termlist self))) (list term))
120 self))
121
122(defun alist->poly (alist &aux (poly (make-instance 'poly)))
123 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
124It can be used to enter simple polynomials by hand, e.g the polynomial
125in two variables, X and Y, given in standard notation as:
126
127 3*X^2*Y^3+2*Y+7
128
129can be entered as
130(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
131
132NOTE: The primary use is for low-level debugging of the package."
133 (dolist (x alist poly)
134 (poly-insert-term poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))
135
136(defmethod update-instance-for-different-class :after ((old term) (new poly) &key)
137 "Converts OLD of class TERM to a NEW of class POLY, by making it into a 1-element TERMLIST."
138 (reinitialize-instance new
139 :dimension (monom-dimension old)
140 :termlist (list old)))
141
142(defmethod update-instance-for-different-class :after ((old monom) (new poly) &key)
143 "Converts OLD of class MONOM to a NEW of class POLY, by making it into a 1-element TERMLIST."
144 (reinitialize-instance new
145 :dimension (monom-dimension old)
146 :termlist (list (change-class old 'term))))
147
148(defmethod universal-equalp ((self poly) (other poly))
149 "Implements equality of polynomials."
150 (and (eql (poly-dimension self) (poly-dimension other))
151 (every #'universal-equalp (poly-termlist self) (poly-termlist other))
152 (eq (poly-term-order self) (poly-term-order other))))
153
154(defgeneric leading-term (object)
155 (:method ((self poly))
156 (car (poly-termlist self)))
157 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
158
159(defgeneric second-leading-term (object)
160 (:method ((self poly))
161 (cadar (poly-termlist self)))
162 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
163
164(defgeneric leading-monomial (object)
165 (:method ((self poly))
166 (change-class (copy-instance (leading-term self)) 'monom))
167 (:documentation "The leading monomial of a polynomial, or NIL for zero polynomial."))
168
169(defgeneric second-leading-monomial (object)
170 (:method ((self poly))
171 (change-class (copy-instance (second-leading-term self)) 'monom))
172 (:documentation "The leading monomial of a polynomial, or NIL for zero polynomial."))
173
174(defgeneric leading-coefficient (object)
175 (:method ((self poly))
176 (term-coeff (leading-term self)))
177 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
178
179(defgeneric second-leading-coefficient (object)
180 (:method ((self poly))
181 (term-coeff (second-leading-term self)))
182 (:documentation "The second leading coefficient of a polynomial. It
183 signals error for a polynomial with at most one term."))
184
185(defmethod universal-zerop ((self poly))
186 "Return T iff SELF is a zero polynomial."
187 (null (poly-termlist self)))
188
189(defgeneric poly-length (self)
190 (:documentation "Return the number of terms.")
191 (:method ((self poly))
192 (length (poly-termlist self))))
193
194(defgeneric scalar-multiply-by (self other)
195 (:documentation "Multiply vector SELF by a scalar OTHER.")
196 (:method ((self poly) other)
197 (mapc #'(lambda (term) (setf (term-coeff term) (multiply (term-coeff term) other)))
198 (poly-termlist self))
199 self))
200
201(defgeneric scalar-divide-by (self other)
202 (:documentation "Divide vector SELF by a scalar OTHER.")
203 (:method ((self poly) other)
204 (mapc #'(lambda (term) (setf (term-coeff term) (divide (term-coeff term) other)))
205 (poly-termlist self))
206 self))
207
208(defmethod unary-inverse :before ((self poly))
209 "Checks invertibility of a polynomial SELF. To be invertable, the
210polynomial must be an invertible, constant polynomial."
211 (with-slots (termlist)
212 self
213 (assert (and (= (length termlist) 1) (zerop (total-degree (car termlist))))
214 nil
215 "To be invertible, the polynomial must have 1 term of total degree 0.")))
216
217(defmethod unary-inverse ((self poly))
218 "Returns the unary inverse of a polynomial SELF."
219 (with-slots (termlist)
220 self
221 (setf (car termlist) (unary-inverse (car termlist)))
222 self))
223
224(defmethod multiply-by ((self poly) (other monom))
225 "Multiply a polynomial SELF by OTHER."
226 (mapc #'(lambda (term) (multiply-by term other))
227 (poly-termlist self))
228 self)
229
230(defmethod multiply-by ((self poly) (other term))
231 "Multiply a polynomial SELF by OTHER."
232 (mapc #'(lambda (term) (multiply-by term other))
233 (poly-termlist self))
234 self)
235
236(defmethod multiply-by ((self monom) (other poly))
237 "Multiply a monomial SELF by polynomial OTHER."
238 (multiply-by other self))
239
240(defmethod multiply-by ((self term) (other poly))
241 "Multiply a term SELF by polynomial OTHER."
242 (multiply-by other self))
243
244(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
245 "Return an expression which will efficiently adds/subtracts two
246polynomials, P and Q. The addition/subtraction of coefficients is
247performed by calling ADD/SUBTRACT-FN. If UMINUS-FN is supplied, it is
248used to negate the coefficients of Q which do not have a corresponding
249coefficient in P. The code implements an efficient algorithm to add
250two polynomials represented as sorted lists of terms. The code
251destroys both arguments, reusing the terms to build the result."
252 `(macrolet ((lc (x) `(term-coeff (car ,x))))
253 (do ((p ,p)
254 (q ,q)
255 r)
256 ((or (endp p) (endp q))
257 ;; NOTE: R contains the result in reverse order. Can it
258 ;; be more efficient to produce the terms in correct order?
259 (unless (endp q)
260 ;; Upon subtraction, we must change the sign of
261 ;; all coefficients in q
262 ,@(when uminus-fn
263 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
264 (setf r (nreconc r q)))
265 (unless (endp p)
266 (setf r (nreconc r p)))
267 r)
268 (multiple-value-bind
269 (greater-p equal-p)
270 (funcall ,order-fn (car p) (car q))
271 (cond
272 (greater-p
273 (rotatef (cdr p) r p)
274 )
275 (equal-p
276 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
277 (cond
278 ((universal-zerop s)
279 (setf p (cdr p))
280 )
281 (t
282 (setf (lc p) s)
283 (rotatef (cdr p) r p))))
284 (setf q (cdr q))
285 )
286 (t
287 ;;Negate the term of Q if UMINUS provided, signallig
288 ;;that we are doing subtraction
289 ,(when uminus-fn
290 `(setf (lc q) (funcall ,uminus-fn (lc q))))
291 (rotatef (cdr q) r q))))
292 ;;(format t "P:~A~%" p)
293 ;;(format t "Q:~A~%" q)
294 ;;(format t "R:~A~%" r)
295 )))
296
297
298
299(defgeneric add-to (self other)
300 (:documentation "Add OTHER to SELF.")
301 (:method ((self number) (other number))
302 (+ self other))
303 (:method ((self poly) (other number))
304 (add-to self (make-poly-constant (poly-dimension self) other)))
305 (:method ((self number) (other poly))
306 (add-to (make-poly-constant (poly-dimension other) self) other)))
307
308
309(defgeneric subtract-from (self other)
310 (:documentation "Subtract OTHER from SELF.")
311 (:method ((self number) (other number))
312 (- self other))
313 (:method ((self poly) (other number))
314 (subtract-from self (make-poly-constant (poly-dimension self) other))))
315
316
317#|
318(defmacro def-add/subtract-method (add/subtract-method-name
319 uminus-method-name
320 &optional
321 (doc-string nil doc-string-supplied-p))
322 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
323 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
324 ,@(when doc-string-supplied-p `(,doc-string))
325 ;; Ensure orders are compatible
326 (change-term-order other self)
327 (setf (poly-termlist self) (fast-add/subtract
328 (poly-termlist self) (poly-termlist other)
329 (poly-term-order self)
330 #',add/subtract-method-name
331 ,(when uminus-method-name `(function ,uminus-method-name))))
332 self))
333
334(eval-when (:load-toplevel :execute)
335
336 (def-add/subtract-method add-to nil
337 "Adds to polynomial SELF another polynomial OTHER.
338This operation destructively modifies both polynomials.
339The result is stored in SELF. This implementation does
340no consing, entirely reusing the sells of SELF and OTHER.")
341
342 (def-add/subtract-method subtract-from unary-minus
343 "Subtracts from polynomial SELF another polynomial OTHER.
344This operation destructively modifies both polynomials.
345The result is stored in SELF. This implementation does
346no consing, entirely reusing the sells of SELF and OTHER.")
347 )
348
349|#
350
351(defmethod unary-minus ((self poly))
352 "Destructively modifies the coefficients of the polynomial SELF,
353by changing their sign."
354 (mapc #'unary-minus (poly-termlist self))
355 self)
356
357(defun add-termlists (p q order-fn)
358 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
359 (fast-add/subtract p q order-fn #'add-to nil))
360
361(defun subtract-termlists (p q order-fn)
362 "Destructively subtracts two termlists P and Q ordered according to ORDER-FN."
363 (fast-add/subtract p q order-fn #'subtract-from #'unary-minus))
364
365(defmethod add-to ((self poly) (other poly))
366 "Adds to polynomial SELF another polynomial OTHER.
367This operation destructively modifies both polynomials.
368The result is stored in SELF. This implementation does
369no consing, entirely reusing the sells of SELF and OTHER."
370 (change-term-order other self)
371 (setf (poly-termlist self) (add-termlists
372 (poly-termlist self) (poly-termlist other)
373 (poly-term-order self)))
374 self)
375
376
377(defmethod subtract-from ((self poly) (other poly))
378 "Subtracts from polynomial SELF another polynomial OTHER.
379This operation destructively modifies both polynomials.
380The result is stored in SELF. This implementation does
381no consing, entirely reusing the sells of SELF and OTHER."
382 (change-term-order other self)
383 (setf (poly-termlist self) (subtract-termlists
384 (poly-termlist self) (poly-termlist other)
385 (poly-term-order self)))
386 self)
387
388
389(defmethod add-to ((self poly) (other term))
390 "Adds to a polynomial SELF a term OTHER. The term OTHER is not
391modified."
392 (add-to self (change-class (copy-instance other) 'poly)))
393
394(defmethod subtract-from ((self poly) (other term))
395 "Subtracts from a polynomial SELF a term OTHER. The term OTHER is not
396modified."
397 (subtract-from self (change-class (copy-instance other) 'poly)))
398
399
400(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
401 &optional (reverse-arg-order-P nil))
402 "Multiplies term TERM by a list of term, TERMLIST.
403Takes into accound divisors of zero in the ring, by
404deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
405is T, change the order of arguments; this may be important
406if we extend the package to non-commutative rings."
407 `(mapcan #'(lambda (other-term)
408 (let ((prod (multiply
409 ,@(cond
410 (reverse-arg-order-p
411 `(other-term ,term))
412 (t
413 `(,term other-term))))))
414 (cond
415 ((universal-zerop prod) nil)
416 (t (list prod)))))
417 ,termlist))
418
419(defun multiply-termlists (p q order-fn)
420 "A version of polynomial multiplication, operating
421directly on termlists."
422 (cond
423 ((or (endp p) (endp q))
424 ;;p or q is 0 (represented by NIL)
425 nil)
426 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
427 ((endp (cdr p))
428 (multiply-term-by-termlist-dropping-zeros (car p) q))
429 ((endp (cdr q))
430 (multiply-term-by-termlist-dropping-zeros (car q) p t))
431 (t
432 (cons (multiply (car p) (car q))
433 (add-termlists
434 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
435 (multiply-termlists (cdr p) q order-fn)
436 order-fn)))))
437
438(defmethod multiply-by ((self poly) (other poly))
439 (change-term-order other self)
440 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
441 (poly-termlist other)
442 (poly-term-order self)))
443 self)
444
445(defgeneric add-2 (object1 object2)
446 (:documentation "Non-destructively add OBJECT1 to OBJECT2.")
447 (:method ((object1 t) (object2 t))
448 (add-to (copy-instance object1) (copy-instance object2))))
449
450(defun add (&rest summands)
451 "Non-destructively adds list SUMMANDS."
452 (cond ((endp summands) 0)
453 (t (reduce #'add-2 summands))))
454
455(defun subtract (minuend &rest subtrahends)
456 "Non-destructively subtract MINUEND and SUBTRAHENDS."
457 (cond ((endp subtrahends) (unary-minus minuend))
458 (t (subtract-from (copy-instance minuend) (reduce #'add subtrahends)))))
459
460(defmethod left-tensor-product-by ((self poly) (other monom))
461 (setf (poly-termlist self)
462 (mapcan #'(lambda (term)
463 (let ((prod (left-tensor-product-by term other)))
464 (cond
465 ((universal-zerop prod) nil)
466 (t (list prod)))))
467 (poly-termlist self)))
468 (incf (poly-dimension self) (monom-dimension other))
469 self)
470
471(defmethod right-tensor-product-by ((self poly) (other monom))
472 (setf (poly-termlist self)
473 (mapcan #'(lambda (term)
474 (let ((prod (right-tensor-product-by term other)))
475 (cond
476 ((universal-zerop prod) nil)
477 (t (list prod)))))
478 (poly-termlist self)))
479 (incf (poly-dimension self) (monom-dimension other))
480 self)
481
482
483(defun standard-extension (plist &aux (k (length plist)) (i 0))
484 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
485is a list of polynomials. Destructively modifies PLIST elements."
486 (mapc #'(lambda (poly)
487 (left-tensor-product-by
488 poly
489 (prog1
490 (make-monom-variable k i)
491 (incf i))))
492 plist))
493
494(defun standard-extension-1 (plist
495 &aux
496 (plist (standard-extension plist))
497 (nvars (poly-dimension (car plist))))
498 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
499Firstly, new K variables U1, U2, ..., UK, are inserted into each
500polynomial. Subsequently, P1, P2, ..., PK are destructively modified
501tantamount to replacing PI with UI*PI-1. It assumes that all
502polynomials have the same dimension, and only the first polynomial
503is examined to determine this dimension."
504 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
505 ;; 1 from each polynomial; since UI*PI has no constant term,
506 ;; we just need to append the constant term at the end
507 ;; of each termlist.
508 (flet ((subtract-1 (p)
509 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
510 (setf plist (mapc #'subtract-1 plist)))
511 plist)
512
513
514(defun standard-sum (plist
515 &aux
516 (plist (standard-extension plist))
517 (nvars (poly-dimension (car plist))))
518 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
519Firstly, new K variables, U1, U2, ..., UK, are inserted into each
520polynomial. Subsequently, P1, P2, ..., PK are destructively modified
521tantamount to replacing PI with UI*PI, and the resulting polynomials
522are added. Finally, 1 is subtracted. It should be noted that the term
523order is not modified, which is equivalent to using a lexicographic
524order on the first K variables."
525 (flet ((subtract-1 (p)
526 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
527 (subtract-1
528 (make-instance
529 'poly
530 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
531
532(defgeneric universal-ezgcd (x y)
533 (:documentation "Solves the diophantine system: X=C*X1, Y=C*X2,
534C=GCD(X,Y). It returns C, X1 and Y1. The result may be obtained by
535the Euclidean algorithm.")
536 (:method ((x integer) (y integer)
537 &aux (c (gcd x y)))
538 (values c (/ x c) (/ y c)))
539 )
540
541(defgeneric s-polynomial (object1 object2)
542 (:documentation "Yields the S-polynomial of OBJECT1 and OBJECT2.")
543 (:method ((f poly) (g poly))
544 (let* ((lcm (universal-lcm (leading-monomial f) (leading-monomial g)))
545 (mf (divide lcm (leading-monomial f)))
546 (mg (divide lcm (leading-monomial g))))
547 (multiple-value-bind (c cf cg)
548 (universal-ezgcd (leading-coefficient f) (leading-coefficient g))
549 (declare (ignore c))
550 (subtract
551 (multiply-by f (change-class mf 'term :coeff cg))
552 (multiply-by g (change-class mg 'term :coeff cf)))))))
553
554(defgeneric poly-content (object)
555 (:documentation "Greatest common divisor of the coefficients of the polynomial object OBJECT.")
556 (:method ((self poly))
557 (reduce #'universal-gcd
558 (mapcar #'term-coeff (rest (poly-termlist self)))
559 :initial-value (leading-coefficient self))))
560
561(defun poly-primitive-part (object)
562 "Divide polynomial OBJECT by gcd of its
563coefficients. Return the resulting polynomial."
564 (scalar-divide-by object (poly-content object)))
565
566(defun poly-insert-variables (self k)
567 (left-tensor-product-by self (make-instance 'monom :dimension k)))
568
569(defun saturation-extension (f plist &aux (k (length plist)))
570 "Calculate [F', U1*P1-1,U2*P2-1,...,UK*PK-1], where
571PLIST=[P1,P2,...,PK] and F' is F with variables U1,U2,...,UK inserted
572as first K variables. It destructively modifies F and PLIST."
573 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
574 (standard-extension-1 plist)))
575
576(defun polysaturation-extension (f plist &aux (k (length plist)))
577 "Calculate [F', U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]
578and F' is F with variables U1,U2,...,UK inserted as first K
579variables. It destructively modifies F and PLIST."
580 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
581 (list (standard-sum plist))))
582
583(defun saturation-extension-1 (f p)
584 "Given family of polynomials F and a polynomial P, calculate [F',
585U*P-1], where F' is F with variable inserted as the first variable. It
586destructively modifies F and P."
587 (polysaturation-extension f (list p)))
588
589(defmethod multiply-by ((object1 number) (object2 poly))
590 (scalar-multiply-by (copy-instance object2) object1))
591
592(defmethod multiply-by ((object1 poly) (object2 number))
593 (scalar-multiply-by (copy-instance object1) object2))
594
595(defun make-poly-variable (nvars pos &optional (power 1))
596 (change-class (make-monom-variable nvars pos power) 'poly))
597
598(defun make-poly-constant (nvars coeff)
599 (change-class (make-term-constant nvars coeff) 'poly))
600
601(defgeneric universal-expt (x y)
602 (:documentation "Raises X to power Y.")
603 (:method ((x number) (y integer)) (expt x y))
604 (:method ((x t) (y integer))
605 (declare (type fixnum y))
606 (cond
607 ((minusp y) (error "universal-expt: Negative exponent."))
608 ((universal-zerop x) (if (zerop y) 1))
609 (t
610 (do ((k 1 (ash k 1))
611 (q x (multiply q q)) ;keep squaring
612 (p 1 (if (not (zerop (logand k y))) (multiply p q) p)))
613 ((> k y) p)
614 (declare (fixnum k)))))))
615
616(defgeneric poly-p (object)
617 (:documentation "Checks if an object is a polynomial.")
618 (:method ((self poly)) t)
619 (:method ((self t)) nil))
620
621(defmethod ->sexp :before ((self poly) &optional vars)
622 "Ensures that the number of variables in VARS maches the polynomial dimension of the
623polynomial SELF."
624 (with-slots (dimension)
625 self
626 (assert (= (length vars) dimension)
627 nil
628 "Number of variables ~S does not match the dimension ~S"
629 vars dimension)))
630
631(defmethod ->sexp ((self poly) &optional vars)
632 "Converts a polynomial SELF to a sexp."
633 (let ((m (mapcar #'(lambda (x) (->sexp x vars))
634 (poly-termlist self))))
635 (cond ((endp m) 0)
636 ((endp (cdr m)) (car m))
637 (t (cons '+ m)))))
638
639(defparameter +list-marker+ :[
640 "A sexp with this head is considered a list of polynomials.")
641
642(defmethod ->sexp ((self cons) &optional vars)
643 (assert (eql (car self) +list-marker+))
644 (cons +list-marker+ (mapcar #'(lambda (p) (->sexp p vars)) (cdr self))))
645
646
647(defun poly-eval (expr vars order)
648 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
649variables VARS. Return the resulting polynomial or list of
650polynomials. Standard arithmetical operators in form EXPR are
651replaced with their analogues in the ring of polynomials, and the
652resulting expression is evaluated, resulting in a polynomial or a list
653of polynomials in internal form. A similar operation in another computer
654algebra system could be called 'expand' or so."
655 (labels ((p-eval (p) (poly-eval p vars order))
656 (p-eval-list (plist) (mapcar #'p-eval plist)))
657 (cond
658 ((eq expr 0)
659 (make-instance 'poly :dimension (length vars)))
660 ((member expr vars :test #'equalp)
661 (let ((pos (position expr vars :test #'equalp)))
662 (make-poly-variable (length vars) pos)))
663 ((atom expr)
664 (make-poly-constant (length vars) expr))
665 ((eq (car expr) +list-marker+)
666 (cons +list-marker+ (p-eval-list (cdr expr))))
667 (t
668 (case (car expr)
669 (+ (reduce #'add (p-eval-list (cdr expr))))
670 (- (apply #'subtract (p-eval-list (cdr expr))))
671 (*
672 (if (endp (cddr expr)) ;unary
673 (p-eval (cadr expr))
674 (apply #'multiply (p-eval-list (cdr expr)))))
675 (/
676 ;; A polynomial can be divided by a scalar
677 (cond
678 ((endp (cddr expr))
679 ;; A special case (/ ?), the inverse
680 (divide (cadr expr)))
681 (t
682 (let ((num (p-eval (cadr expr)))
683 (denom-inverse (apply #'divide (mapcar #'p-eval (cddr expr)))))
684 (multiply denom-inverse num)))))
685 (expt
686 (cond
687 ((member (cadr expr) vars :test #'equalp)
688 ;;Special handling of (expt var pow)
689 (let ((pos (position (cadr expr) vars :test #'equalp)))
690 (make-poly-variable (length vars) pos (caddr expr))))
691 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
692 ;; Negative power means division in coefficient ring
693 ;; Non-integer power means non-polynomial coefficient
694 expr)
695 (t (universal-expt (p-eval (cadr expr)) (caddr expr)))))
696 (otherwise
697 (error "Cannot evaluate as polynomial: ~A" expr)))))))
698
699(defgeneric make-zero-for (self)
700 (:method ((self poly))
701 (make-instance 'poly :dimension (poly-dimension self))))
702
703(defgeneric make-unit-for (self)
704 (:method ((self poly))
705 (make-poly-constant (poly-dimension self) 1)))
706
707(defgeneric poly-reverse (self)
708 (:documentation "Reverse the order of terms in a polynomial SELF.")
709 (:method ((self poly))
710 (with-slots (termlist)
711 self
712 (setf termlist (nreverse termlist)))
713 self))
714
715
716
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