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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 "UNIVERSAL-EXPT"
55 "POLY-P")
56 (:documentation "Implements polynomials. A polynomial is essentially
57a mapping of monomials of the same degree to coefficients. The
58momomials are ordered according to a monomial order."))
59
60(in-package :polynomial)
61
62(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))
63
64(defclass poly ()
65 ((dimension :initform nil
66 :initarg :dimension
67 :accessor poly-dimension
68 :documentation "Shared dimension of all terms, the number of variables")
69 (termlist :initform nil :initarg :termlist :accessor poly-termlist
70 :documentation "List of terms.")
71 (order :initform #'lex> :initarg :order :accessor poly-term-order
72 :documentation "Monomial/term order."))
73 (:default-initargs :dimension nil :termlist nil :order #'lex>)
74 (:documentation "A polynomial with a list of terms TERMLIST, ordered
75according to term order ORDER, which defaults to LEX>."))
76
77(defmethod print-object ((self poly) stream)
78 (print-unreadable-object (self stream :type t :identity t)
79 (with-accessors ((dimension poly-dimension)
80 (termlist poly-termlist)
81 (order poly-term-order))
82 self
83 (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A"
84 dimension termlist order))))
85
86(defgeneric change-term-order (self other)
87 (:documentation "Change term order of SELF to the term order of OTHER.")
88 (:method ((self poly) (other poly))
89 (unless (eq (poly-term-order self) (poly-term-order other))
90 (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))
91 (poly-term-order self) (poly-term-order other)))
92 self))
93
94(defgeneric poly-insert-term (self term)
95 (:documentation "Insert a term TERM into SELF before all other
96 terms. Order is not enforced.")
97 (:method ((self poly) (term term))
98 (cond ((null (poly-dimension self))
99 (setf (poly-dimension self) (monom-dimension term)))
100 (t (assert (= (poly-dimension self) (monom-dimension term)))))
101 (push term (poly-termlist self))
102 self))
103
104(defgeneric poly-append-term (self term)
105 (:documentation "Append a term TERM to SELF after all other 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 (setf (cdr (last (poly-termlist self))) (list term))
111 self))
112
113(defun alist->poly (alist &aux (poly (make-instance 'poly)))
114 "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).
115It can be used to enter simple polynomials by hand, e.g the polynomial
116in two variables, X and Y, given in standard notation as:
117
118 3*X^2*Y^3+2*Y+7
119
120can be entered as
121(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).
122
123NOTE: The primary use is for low-level debugging of the package."
124 (dolist (x alist poly)
125 (poly-insert-term poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))
126
127(defmethod update-instance-for-different-class :after ((old term) (new poly) &key)
128 "Converts OLD of class TERM to a NEW of class POLY, by making it into a 1-element TERMLIST."
129 (reinitialize-instance new
130 :dimension (monom-dimension old)
131 :termlist (list old)))
132
133(defmethod universal-equalp ((self poly) (other poly))
134 "Implements equality of polynomials."
135 (and (eql (poly-dimension self) (poly-dimension other))
136 (every #'universal-equalp (poly-termlist self) (poly-termlist other))
137 (eq (poly-term-order self) (poly-term-order other))))
138
139(defgeneric leading-term (object)
140 (:method ((self poly))
141 (car (poly-termlist self)))
142 (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))
143
144(defgeneric second-leading-term (object)
145 (:method ((self poly))
146 (cadar (poly-termlist self)))
147 (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))
148
149(defgeneric leading-monomial (object)
150 (:method ((self poly))
151 (change-class (copy-instance (leading-term self)) 'monom))
152 (:documentation "The leading monomial of a polynomial, or NIL for zero polynomial."))
153
154(defgeneric second-leading-monomial (object)
155 (:method ((self poly))
156 (change-class (copy-instance (second-leading-term self)) 'monom))
157 (:documentation "The leading monomial of a polynomial, or NIL for zero polynomial."))
158
159(defgeneric leading-coefficient (object)
160 (:method ((self poly))
161 (term-coeff (leading-term self)))
162 (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))
163
164(defgeneric second-leading-coefficient (object)
165 (:method ((self poly))
166 (term-coeff (second-leading-term self)))
167 (:documentation "The second leading coefficient of a polynomial. It
168 signals error for a polynomial with at most one term."))
169
170(defmethod universal-zerop ((self poly))
171 "Return T iff SELF is a zero polynomial."
172 (null (poly-termlist self)))
173
174(defgeneric poly-length (self)
175 (:documentation "Return the number of terms.")
176 (:method ((self poly))
177 (length (poly-termlist self))))
178
179(defgeneric scalar-multiply-by (self other)
180 (:documentation "Multiply vector SELF by a scalar OTHER.")
181 (:method ((self poly) other)
182 (mapc #'(lambda (term) (setf (term-coeff term) (multiply (term-coeff term) other)))
183 (poly-termlist self))
184 self))
185
186(defgeneric scalar-divide-by (self other)
187 (:documentation "Divide vector SELF by a scalar OTHER.")
188 (:method ((self poly) other)
189 (mapc #'(lambda (term) (setf (term-coeff term) (divide (term-coeff term) other)))
190 (poly-termlist self))
191 self))
192
193(defmethod multiply-by ((self poly) (other monom))
194 "Multiply a polynomial SELF by OTHER."
195 (mapc #'(lambda (term) (multiply-by term other))
196 (poly-termlist self))
197 self)
198
199(defmethod multiply-by ((self poly) (other term))
200 "Multiply a polynomial SELF by OTHER."
201 (mapc #'(lambda (term) (multiply-by term other))
202 (poly-termlist self))
203 self)
204
205(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
206 "Return an expression which will efficiently adds/subtracts two
207polynomials, P and Q. The addition/subtraction of coefficients is
208performed by calling ADD/SUBTRACT-METHOD-NAME. If UMINUS-METHOD-NAME
209is supplied, it is used to negate the coefficients of Q which do not
210have a corresponding coefficient in P. The code implements an
211efficient algorithm to add two polynomials represented as sorted lists
212of terms. The code destroys both arguments, reusing the terms to build
213the result."
214 `(macrolet ((lc (x) `(term-coeff (car ,x))))
215 (do ((p ,p)
216 (q ,q)
217 r)
218 ((or (endp p) (endp q))
219 ;; NOTE: R contains the result in reverse order. Can it
220 ;; be more efficient to produce the terms in correct order?
221 (unless (endp q)
222 ;; Upon subtraction, we must change the sign of
223 ;; all coefficients in q
224 ,@(when uminus-fn
225 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
226 (setf r (nreconc r q)))
227 r)
228 (multiple-value-bind
229 (greater-p equal-p)
230 (funcall ,order-fn (car p) (car q))
231 (cond
232 (greater-p
233 (rotatef (cdr p) r p)
234 )
235 (equal-p
236 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
237 (cond
238 ((universal-zerop s)
239 (setf p (cdr p))
240 )
241 (t
242 (setf (lc p) s)
243 (rotatef (cdr p) r p))))
244 (setf q (cdr q))
245 )
246 (t
247 ;;Negate the term of Q if UMINUS provided, signallig
248 ;;that we are doing subtraction
249 ,(when uminus-fn
250 `(setf (lc q) (funcall ,uminus-fn (lc q))))
251 (rotatef (cdr q) r q)))))))
252
253
254(defgeneric add-to (self other)
255 (:documentation "Add OTHER to SELF.")
256 (:method ((self number) (other number))
257 (+ self other)))
258
259(defgeneric subtract-from (self other)
260 (:documentation "Subtract OTHER from SELF.")
261 (:method ((self number) (other number))
262 (- self other)))
263
264(defmacro def-add/subtract-method (add/subtract-method-name
265 uminus-method-name
266 &optional
267 (doc-string nil doc-string-supplied-p))
268 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
269 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
270 ,@(when doc-string-supplied-p `(,doc-string))
271 ;; Ensure orders are compatible
272 (change-term-order other self)
273 (setf (poly-termlist self) (fast-add/subtract
274 (poly-termlist self) (poly-termlist other)
275 (poly-term-order self)
276 #',add/subtract-method-name
277 ,(when uminus-method-name `(function ,uminus-method-name))))
278 self))
279
280(eval-when (:load-toplevel :execute)
281
282 (def-add/subtract-method add-to nil
283 "Adds to polynomial SELF another polynomial OTHER.
284This operation destructively modifies both polynomials.
285The result is stored in SELF. This implementation does
286no consing, entirely reusing the sells of SELF and OTHER.")
287
288 (def-add/subtract-method subtract-from unary-minus
289 "Subtracts from polynomial SELF another polynomial OTHER.
290This operation destructively modifies both polynomials.
291The result is stored in SELF. This implementation does
292no consing, entirely reusing the sells of SELF and OTHER.")
293 )
294
295(defmethod unary-minus ((self poly))
296 "Destructively modifies the coefficients of the polynomial SELF,
297by changing their sign."
298 (mapc #'unary-minus (poly-termlist self))
299 self)
300
301(defun add-termlists (p q order-fn)
302 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
303 (fast-add/subtract p q order-fn #'add-to nil))
304
305(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
306 &optional (reverse-arg-order-P nil))
307 "Multiplies term TERM by a list of term, TERMLIST.
308Takes into accound divisors of zero in the ring, by
309deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
310is T, change the order of arguments; this may be important
311if we extend the package to non-commutative rings."
312 `(mapcan #'(lambda (other-term)
313 (let ((prod (multiply
314 ,@(cond
315 (reverse-arg-order-p
316 `(other-term ,term))
317 (t
318 `(,term other-term))))))
319 (cond
320 ((universal-zerop prod) nil)
321 (t (list prod)))))
322 ,termlist))
323
324(defun multiply-termlists (p q order-fn)
325 "A version of polynomial multiplication, operating
326directly on termlists."
327 (cond
328 ((or (endp p) (endp q))
329 ;;p or q is 0 (represented by NIL)
330 nil)
331 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
332 ((endp (cdr p))
333 (multiply-term-by-termlist-dropping-zeros (car p) q))
334 ((endp (cdr q))
335 (multiply-term-by-termlist-dropping-zeros (car q) p t))
336 (t
337 (cons (multiply (car p) (car q))
338 (add-termlists
339 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
340 (multiply-termlists (cdr p) q order-fn)
341 order-fn)))))
342
343(defmethod multiply-by ((self poly) (other poly))
344 (change-term-order other self)
345 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
346 (poly-termlist other)
347 (poly-term-order self)))
348 self)
349
350(defun add (object1 object2)
351 "Non-destructively add POLY1 by POLY2."
352 (add-to (copy-instance object1) (copy-instance object2)))
353
354(defun subtract (minuend &rest subtrahends)
355 "Non-destructively subtract MINUEND and SUBTRAHENDS."
356 (subtract-from (copy-instance minuend) (reduce #'add subtrahends)))
357
358(defmethod left-tensor-product-by ((self poly) (other monom))
359 (setf (poly-termlist self)
360 (mapcan #'(lambda (term)
361 (let ((prod (left-tensor-product-by term other)))
362 (cond
363 ((universal-zerop prod) nil)
364 (t (list prod)))))
365 (poly-termlist self)))
366 (incf (poly-dimension self) (monom-dimension other))
367 self)
368
369(defmethod right-tensor-product-by ((self poly) (other monom))
370 (setf (poly-termlist self)
371 (mapcan #'(lambda (term)
372 (let ((prod (right-tensor-product-by term other)))
373 (cond
374 ((universal-zerop prod) nil)
375 (t (list prod)))))
376 (poly-termlist self)))
377 (incf (poly-dimension self) (monom-dimension other))
378 self)
379
380
381(defun standard-extension (plist &aux (k (length plist)) (i 0))
382 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
383is a list of polynomials. Destructively modifies PLIST elements."
384 (mapc #'(lambda (poly)
385 (left-tensor-product-by
386 poly
387 (prog1
388 (make-monom-variable k i)
389 (incf i))))
390 plist))
391
392(defun standard-extension-1 (plist
393 &aux
394 (plist (standard-extension plist))
395 (nvars (poly-dimension (car plist))))
396 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
397Firstly, new K variables U1, U2, ..., UK, are inserted into each
398polynomial. Subsequently, P1, P2, ..., PK are destructively modified
399tantamount to replacing PI with UI*PI-1. It assumes that all
400polynomials have the same dimension, and only the first polynomial
401is examined to determine this dimension."
402 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
403 ;; 1 from each polynomial; since UI*PI has no constant term,
404 ;; we just need to append the constant term at the end
405 ;; of each termlist.
406 (flet ((subtract-1 (p)
407 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
408 (setf plist (mapc #'subtract-1 plist)))
409 plist)
410
411
412(defun standard-sum (plist
413 &aux
414 (plist (standard-extension plist))
415 (nvars (poly-dimension (car plist))))
416 "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].
417Firstly, new K variables, U1, U2, ..., UK, are inserted into each
418polynomial. Subsequently, P1, P2, ..., PK are destructively modified
419tantamount to replacing PI with UI*PI, and the resulting polynomials
420are added. Finally, 1 is subtracted. It should be noted that the term
421order is not modified, which is equivalent to using a lexicographic
422order on the first K variables."
423 (flet ((subtract-1 (p)
424 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
425 (subtract-1
426 (make-instance
427 'poly
428 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
429
430(defgeneric universal-ezgcd (x y)
431 (:documentation "Solves the diophantine system: X=C*X1, Y=C*X2,
432C=GCD(X,Y). It returns C, X1 and Y1. The result may be obtained by
433the Euclidean algorithm.")
434 (:method ((x integer) (y integer)
435 &aux (c (gcd x y)))
436 (values c (/ x c) (/ y c)))
437 )
438
439(defgeneric s-polynomial (object1 object2)
440 (:documentation "Yields the S-polynomial of OBJECT1 and OBJECT2.")
441 (:method ((f poly) (g poly))
442 (let* ((lcm (universal-lcm (leading-monomial f) (leading-monomial g)))
443 (mf (divide lcm (leading-monomial f)))
444 (mg (divide lcm (leading-monomial g))))
445 (multiple-value-bind (c cf cg)
446 (universal-ezgcd (leading-coefficient f) (leading-coefficient g))
447 (declare (ignore c))
448 (subtract
449 (multiply f (change-class mf 'term :coeff cg))
450 (multiply g (change-class mg 'term :coeff cf)))))))
451
452(defgeneric poly-content (object)
453 (:documentation "Greatest common divisor of the coefficients of the polynomial object OBJECT.")
454 (:method ((self poly))
455 (reduce #'universal-gcd
456 (mapcar #'term-coeff (rest (poly-termlist self)))
457 :initial-value (leading-coefficient self))))
458
459(defun poly-primitive-part (object)
460 "Divide polynomial OBJECT by gcd of its
461coefficients. Return the resulting polynomial."
462 (scalar-divide-by object (poly-content object)))
463
464(defun poly-insert-variables (self k)
465 (left-tensor-product-by self (make-instance 'monom :dimension k)))
466
467(defun saturation-extension (f plist &aux (k (length plist)))
468 "Calculate [F', U1*P1-1,U2*P2-1,...,UK*PK-1], where
469PLIST=[P1,P2,...,PK] and F' is F with variables U1,U2,...,UK inserted
470as first K variables. It destructively modifies F and PLIST."
471 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
472 (standard-extension-1 plist)))
473
474(defun polysaturation-extension (f plist &aux (k (length plist)))
475 "Calculate [F', U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]
476and F' is F with variables U1,U2,...,UK inserted as first K
477variables. It destructively modifies F and PLIST."
478 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
479 (list (standard-sum plist))))
480
481(defun saturation-extension-1 (f p)
482 "Given family of polynomials F and a polynomial P, calculate [F',
483U*P-1], where F' is F with variable inserted as the first variable. It
484destructively modifies F and P."
485 (polysaturation-extension f (list p)))
486
487(defmethod multiply-by ((object1 number) (object2 poly))
488 (scalar-multiply-by (copy-instance object2) object1))
489
490(defun make-poly-variable (nvars pos &optional (power 1))
491 (change-class (make-monom-variable nvars pos power) 'poly))
492
493(defgeneric universal-expt (x y)
494 (:documentation "Raises X to power Y.")
495 (:method ((x number) (y integer)) (expt x y))
496 (:method ((x t) (y integer))
497 (declare (type fixnum y))
498 (cond
499 ((minusp y) (error "universal-expt: Negative exponent."))
500 ((universal-zerop x) (if (zerop y) 1))
501 (t
502 (do ((k 1 (ash k 1))
503 (q x (multiply q q)) ;keep squaring
504 (p 1 (if (not (zerop (logand k y))) (multiply p q) p)))
505 ((> k y) p)
506 (declare (fixnum k)))))))
507
508(defgeneric poly-p (object)
509 (:documentation "Checks if an object is a polynomial.")
510 (:method ((self poly)) t)
511 (:method ((self t)) nil))
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