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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 "UNIVERSAL-EZGCD"
49 "S-POLYNOMIAL"
50 "POLY-CONTENT"
51 "POLY-PRIMITIVE-PART"
52 "SATURATION-EXTENSION-1"
53 "MAKE-POLY-VARIABLE"
54 "MAKE-POLY-CONSTANT"
55 "UNIVERSAL-EXPT"
56 "POLY-P")
57 (:documentation "Implements polynomials. A polynomial is essentially
58a mapping of monomials of the same degree to coefficients. The
59momomials are ordered according to a monomial order."))
60
61(in-package :polynomial)
62
63(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
64
65(defclass poly ()
66 ((dimension :initform nil
67 :initarg :dimension
68 :accessor poly-dimension
69 :documentation "Shared dimension of all terms, the number of variables")
70 (termlist :initform nil :initarg :termlist :accessor poly-termlist
71 :documentation "List of terms.")
72 (order :initform #'lex> :initarg :order :accessor poly-term-order
73 :documentation "Monomial/term order."))
74 (:default-initargs :dimension nil :termlist nil :order #'lex>)
75 (:documentation "A polynomial with a list of terms TERMLIST, ordered
76according to term order ORDER, which defaults to LEX>."))
77
78(defmethod print-object ((self poly) stream)
79 (print-unreadable-object (self stream :type t :identity t)
80 (with-accessors ((dimension poly-dimension)
81 (termlist poly-termlist)
82 (order poly-term-order))
83 self
84 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
85 dimension termlist order))))
86
87(defgeneric change-term-order (self other)
88 (:documentation "Change term order of SELF to the term order of OTHER.")
89 (:method ((self poly) (other poly))
90 (unless (eq (poly-term-order self) (poly-term-order other))
91 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
92 (poly-term-order self) (poly-term-order other)))
93 self))
94
95(defgeneric poly-insert-term (self term)
96 (:documentation "Insert a term TERM into SELF before all other
97 terms. Order is not enforced.")
98 (:method ((self poly) (term term))
99 (cond ((null (poly-dimension self))
100 (setf (poly-dimension self) (monom-dimension term)))
101 (t (assert (= (poly-dimension self) (monom-dimension term)))))
102 (push term (poly-termlist self))
103 self))
104
105(defgeneric poly-append-term (self term)
106 (:documentation "Append a term TERM to SELF after all other terms. Order is not enforced.")
107 (:method ((self poly) (term term))
108 (cond ((null (poly-dimension self))
109 (setf (poly-dimension self) (monom-dimension term)))
110 (t (assert (= (poly-dimension self) (monom-dimension term)))))
111 (setf (cdr (last (poly-termlist self))) (list term))
112 self))
113
114(defun alist->poly (alist &aux (poly (make-instance 'poly)))
115 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
116It can be used to enter simple polynomials by hand, e.g the polynomial
117in two variables, X and Y, given in standard notation as:
118
119 3*X^2*Y^3+2*Y+7
120
121can be entered as
122(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
123
124NOTE: The primary use is for low-level debugging of the package."
125 (dolist (x alist poly)
126 (poly-insert-term poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))
127
128(defmethod update-instance-for-different-class :before ((old term) (new poly) &key)
129 "Converts OLD of class TERM to a NEW of class POLY, by making it into a 1-element TERMLIST."
130 (reinitialize-instance new
131 :dimension (monom-dimension old)
132 :termlist (list old)))
133
134(defmethod update-instance-for-different-class :before ((old monom) (new poly) &key)
135 "Converts OLD of class MONOM to a NEW of class POLY, by making it into a 1-element TERMLIST."
136 (reinitialize-instance new
137 :dimension (monom-dimension old)
138 :termlist (list (change-class old 'term))))
139
140(defmethod universal-equalp ((self poly) (other poly))
141 "Implements equality of polynomials."
142 (and (eql (poly-dimension self) (poly-dimension other))
143 (every #'universal-equalp (poly-termlist self) (poly-termlist other))
144 (eq (poly-term-order self) (poly-term-order other))))
145
146(defgeneric leading-term (object)
147 (:method ((self poly))
148 (car (poly-termlist self)))
149 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
150
151(defgeneric second-leading-term (object)
152 (:method ((self poly))
153 (cadar (poly-termlist self)))
154 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
155
156(defgeneric leading-monomial (object)
157 (:method ((self poly))
158 (change-class (copy-instance (leading-term self)) 'monom))
159 (:documentation "The leading monomial of a polynomial, or NIL for zero polynomial."))
160
161(defgeneric second-leading-monomial (object)
162 (:method ((self poly))
163 (change-class (copy-instance (second-leading-term self)) 'monom))
164 (:documentation "The leading monomial of a polynomial, or NIL for zero polynomial."))
165
166(defgeneric leading-coefficient (object)
167 (:method ((self poly))
168 (term-coeff (leading-term self)))
169 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
170
171(defgeneric second-leading-coefficient (object)
172 (:method ((self poly))
173 (term-coeff (second-leading-term self)))
174 (:documentation "The second leading coefficient of a polynomial. It
175 signals error for a polynomial with at most one term."))
176
177(defmethod universal-zerop ((self poly))
178 "Return T iff SELF is a zero polynomial."
179 (null (poly-termlist self)))
180
181(defgeneric poly-length (self)
182 (:documentation "Return the number of terms.")
183 (:method ((self poly))
184 (length (poly-termlist self))))
185
186(defgeneric scalar-multiply-by (self other)
187 (:documentation "Multiply vector SELF by a scalar OTHER.")
188 (:method ((self poly) other)
189 (mapc #'(lambda (term) (setf (term-coeff term) (multiply (term-coeff term) other)))
190 (poly-termlist self))
191 self))
192
193(defgeneric scalar-divide-by (self other)
194 (:documentation "Divide vector SELF by a scalar OTHER.")
195 (:method ((self poly) other)
196 (mapc #'(lambda (term) (setf (term-coeff term) (divide (term-coeff term) other)))
197 (poly-termlist self))
198 self))
199
200(defmethod multiply-by ((self poly) (other monom))
201 "Multiply a polynomial SELF by OTHER."
202 (mapc #'(lambda (term) (multiply-by term other))
203 (poly-termlist self))
204 self)
205
206(defmethod multiply-by ((self poly) (other term))
207 "Multiply a polynomial SELF by OTHER."
208 (mapc #'(lambda (term) (multiply-by term other))
209 (poly-termlist self))
210 self)
211
212(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
213 "Return an expression which will efficiently adds/subtracts two
214polynomials, P and Q. The addition/subtraction of coefficients is
215performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
216is supplied, it is used to negate the coefficients of Q which do not
217have a corresponding coefficient in P. The code implements an
218efficient algorithm to add two polynomials represented as sorted lists
219of terms. The code destroys both arguments, reusing the terms to build
220the result."
221 `(macrolet ((lc (x) `(term-coeff (car ,x))))
222 (do ((p ,p)
223 (q ,q)
224 r)
225 ((or (endp p) (endp q))
226 ;; NOTE: R contains the result in reverse order. Can it
227 ;; be more efficient to produce the terms in correct order?
228 (unless (endp q)
229 ;; Upon subtraction, we must change the sign of
230 ;; all coefficients in q
231 ,@(when uminus-fn
232 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
233 (setf r (nreconc r q)))
234 r)
235 (multiple-value-bind
236 (greater-p equal-p)
237 (funcall ,order-fn (car p) (car q))
238 (cond
239 (greater-p
240 (rotatef (cdr p) r p)
241 )
242 (equal-p
243 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
244 (cond
245 ((universal-zerop s)
246 (setf p (cdr p))
247 )
248 (t
249 (setf (lc p) s)
250 (rotatef (cdr p) r p))))
251 (setf q (cdr q))
252 )
253 (t
254 ;;Negate the term of Q if UMINUS provided, signallig
255 ;;that we are doing subtraction
256 ,(when uminus-fn
257 `(setf (lc q) (funcall ,uminus-fn (lc q))))
258 (rotatef (cdr q) r q)))))))
259
260
261(defgeneric add-to (self other)
262 (:documentation "Add OTHER to SELF.")
263 (:method ((self number) (other number))
264 (+ self other))
265 (:method ((self poly) (other number))
266 (add-to self (make-poly-constant (poly-dimension self) other))))
267
268
269(defgeneric subtract-from (self other)
270 (:documentation "Subtract OTHER from SELF.")
271 (:method ((self number) (other number))
272 (- self other)))
273
274(defmacro def-add/subtract-method (add/subtract-method-name
275 uminus-method-name
276 &optional
277 (doc-string nil doc-string-supplied-p))
278 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
279 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
280 ,@(when doc-string-supplied-p `(,doc-string))
281 ;; Ensure orders are compatible
282 (change-term-order other self)
283 (setf (poly-termlist self) (fast-add/subtract
284 (poly-termlist self) (poly-termlist other)
285 (poly-term-order self)
286 #',add/subtract-method-name
287 ,(when uminus-method-name `(function ,uminus-method-name))))
288 self))
289
290(eval-when (:load-toplevel :execute)
291
292 (def-add/subtract-method add-to nil
293 "Adds to polynomial SELF another polynomial OTHER.
294This operation destructively modifies both polynomials.
295The result is stored in SELF. This implementation does
296no consing, entirely reusing the sells of SELF and OTHER.")
297
298 (def-add/subtract-method subtract-from unary-minus
299 "Subtracts from polynomial SELF another polynomial OTHER.
300This operation destructively modifies both polynomials.
301The result is stored in SELF. This implementation does
302no consing, entirely reusing the sells of SELF and OTHER.")
303 )
304
305(defmethod unary-minus ((self poly))
306 "Destructively modifies the coefficients of the polynomial SELF,
307by changing their sign."
308 (mapc #'unary-minus (poly-termlist self))
309 self)
310
311(defun add-termlists (p q order-fn)
312 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
313 (fast-add/subtract p q order-fn #'add-to nil))
314
315(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
316 &optional (reverse-arg-order-P nil))
317 "Multiplies term TERM by a list of term, TERMLIST.
318Takes into accound divisors of zero in the ring, by
319deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
320is T, change the order of arguments; this may be important
321if we extend the package to non-commutative rings."
322 `(mapcan #'(lambda (other-term)
323 (let ((prod (multiply
324 ,@(cond
325 (reverse-arg-order-p
326 `(other-term ,term))
327 (t
328 `(,term other-term))))))
329 (cond
330 ((universal-zerop prod) nil)
331 (t (list prod)))))
332 ,termlist))
333
334(defun multiply-termlists (p q order-fn)
335 "A version of polynomial multiplication, operating
336directly on termlists."
337 (cond
338 ((or (endp p) (endp q))
339 ;;p or q is 0 (represented by NIL)
340 nil)
341 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
342 ((endp (cdr p))
343 (multiply-term-by-termlist-dropping-zeros (car p) q))
344 ((endp (cdr q))
345 (multiply-term-by-termlist-dropping-zeros (car q) p t))
346 (t
347 (cons (multiply (car p) (car q))
348 (add-termlists
349 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
350 (multiply-termlists (cdr p) q order-fn)
351 order-fn)))))
352
353(defmethod multiply-by ((self poly) (other poly))
354 (change-term-order other self)
355 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
356 (poly-termlist other)
357 (poly-term-order self)))
358 self)
359
360(defgeneric add-2 (object1 object2)
361 (:documentation "Non-destructively add OBJECT1 to OBJECT2.")
362 (:method ((object1 t) (object2 t))
363 (add-to (copy-instance object1) (copy-instance object2))))
364
365(defun add (&rest summands)
366 "Non-destructively adds list SUMMANDS."
367 (cond ((endp summands) 0)
368 (t (reduce #'add-2 summands))))
369
370(defun subtract (minuend &rest subtrahends)
371 "Non-destructively subtract MINUEND and SUBTRAHENDS."
372 (subtract-from (copy-instance minuend) (reduce #'add subtrahends)))
373
374(defmethod left-tensor-product-by ((self poly) (other monom))
375 (setf (poly-termlist self)
376 (mapcan #'(lambda (term)
377 (let ((prod (left-tensor-product-by term other)))
378 (cond
379 ((universal-zerop prod) nil)
380 (t (list prod)))))
381 (poly-termlist self)))
382 (incf (poly-dimension self) (monom-dimension other))
383 self)
384
385(defmethod right-tensor-product-by ((self poly) (other monom))
386 (setf (poly-termlist self)
387 (mapcan #'(lambda (term)
388 (let ((prod (right-tensor-product-by term other)))
389 (cond
390 ((universal-zerop prod) nil)
391 (t (list prod)))))
392 (poly-termlist self)))
393 (incf (poly-dimension self) (monom-dimension other))
394 self)
395
396
397(defun standard-extension (plist &aux (k (length plist)) (i 0))
398 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
399is a list of polynomials. Destructively modifies PLIST elements."
400 (mapc #'(lambda (poly)
401 (left-tensor-product-by
402 poly
403 (prog1
404 (make-monom-variable k i)
405 (incf i))))
406 plist))
407
408(defun standard-extension-1 (plist
409 &aux
410 (plist (standard-extension plist))
411 (nvars (poly-dimension (car plist))))
412 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
413Firstly, new K variables U1, U2, ..., UK, are inserted into each
414polynomial. Subsequently, P1, P2, ..., PK are destructively modified
415tantamount to replacing PI with UI*PI-1. It assumes that all
416polynomials have the same dimension, and only the first polynomial
417is examined to determine this dimension."
418 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
419 ;; 1 from each polynomial; since UI*PI has no constant term,
420 ;; we just need to append the constant term at the end
421 ;; of each termlist.
422 (flet ((subtract-1 (p)
423 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
424 (setf plist (mapc #'subtract-1 plist)))
425 plist)
426
427
428(defun standard-sum (plist
429 &aux
430 (plist (standard-extension plist))
431 (nvars (poly-dimension (car plist))))
432 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
433Firstly, new K variables, U1, U2, ..., UK, are inserted into each
434polynomial. Subsequently, P1, P2, ..., PK are destructively modified
435tantamount to replacing PI with UI*PI, and the resulting polynomials
436are added. Finally, 1 is subtracted. It should be noted that the term
437order is not modified, which is equivalent to using a lexicographic
438order on the first K variables."
439 (flet ((subtract-1 (p)
440 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
441 (subtract-1
442 (make-instance
443 'poly
444 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
445
446(defgeneric universal-ezgcd (x y)
447 (:documentation "Solves the diophantine system: X=C*X1, Y=C*X2,
448C=GCD(X,Y). It returns C, X1 and Y1. The result may be obtained by
449the Euclidean algorithm.")
450 (:method ((x integer) (y integer)
451 &aux (c (gcd x y)))
452 (values c (/ x c) (/ y c)))
453 )
454
455(defgeneric s-polynomial (object1 object2)
456 (:documentation "Yields the S-polynomial of OBJECT1 and OBJECT2.")
457 (:method ((f poly) (g poly))
458 (let* ((lcm (universal-lcm (leading-monomial f) (leading-monomial g)))
459 (mf (divide lcm (leading-monomial f)))
460 (mg (divide lcm (leading-monomial g))))
461 (multiple-value-bind (c cf cg)
462 (universal-ezgcd (leading-coefficient f) (leading-coefficient g))
463 (declare (ignore c))
464 (subtract
465 (multiply f (change-class mf 'term :coeff cg))
466 (multiply g (change-class mg 'term :coeff cf)))))))
467
468(defgeneric poly-content (object)
469 (:documentation "Greatest common divisor of the coefficients of the polynomial object OBJECT.")
470 (:method ((self poly))
471 (reduce #'universal-gcd
472 (mapcar #'term-coeff (rest (poly-termlist self)))
473 :initial-value (leading-coefficient self))))
474
475(defun poly-primitive-part (object)
476 "Divide polynomial OBJECT by gcd of its
477coefficients. Return the resulting polynomial."
478 (scalar-divide-by object (poly-content object)))
479
480(defun poly-insert-variables (self k)
481 (left-tensor-product-by self (make-instance 'monom :dimension k)))
482
483(defun saturation-extension (f plist &aux (k (length plist)))
484 "Calculate [F', U1*P1-1,U2*P2-1,...,UK*PK-1], where
485PLIST=[P1,P2,...,PK] and F' is F with variables U1,U2,...,UK inserted
486as first K variables. It destructively modifies F and PLIST."
487 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
488 (standard-extension-1 plist)))
489
490(defun polysaturation-extension (f plist &aux (k (length plist)))
491 "Calculate [F', U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]
492and F' is F with variables U1,U2,...,UK inserted as first K
493variables. It destructively modifies F and PLIST."
494 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
495 (list (standard-sum plist))))
496
497(defun saturation-extension-1 (f p)
498 "Given family of polynomials F and a polynomial P, calculate [F',
499U*P-1], where F' is F with variable inserted as the first variable. It
500destructively modifies F and P."
501 (polysaturation-extension f (list p)))
502
503(defmethod multiply-by ((object1 number) (object2 poly))
504 (scalar-multiply-by (copy-instance object2) object1))
505
506(defun make-poly-variable (nvars pos &optional (power 1))
507 (change-class (make-monom-variable nvars pos power) 'poly))
508
509(defun make-poly-constant (nvars coeff)
510 (change-class (make-term-constant nvars coeff) 'poly))
511
512(defgeneric universal-expt (x y)
513 (:documentation "Raises X to power Y.")
514 (:method ((x number) (y integer)) (expt x y))
515 (:method ((x t) (y integer))
516 (declare (type fixnum y))
517 (cond
518 ((minusp y) (error "universal-expt: Negative exponent."))
519 ((universal-zerop x) (if (zerop y) 1))
520 (t
521 (do ((k 1 (ash k 1))
522 (q x (multiply q q)) ;keep squaring
523 (p 1 (if (not (zerop (logand k y))) (multiply p q) p)))
524 ((> k y) p)
525 (declare (fixnum k)))))))
526
527(defgeneric poly-p (object)
528 (:documentation "Checks if an object is a polynomial.")
529 (:method ((self poly)) t)
530 (:method ((self t)) nil))
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