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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(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)
237 "Return an expression which will efficiently adds/subtracts two
238polynomials, P and Q. The addition/subtraction of coefficients is
239performed by calling ADD/SUBTRACT-FN. If UMINUS-FN is supplied, it is
240used to negate the coefficients of Q which do not have a corresponding
241coefficient in P. The code implements an efficient algorithm to add
242two polynomials represented as sorted lists of terms. The code
243destroys both arguments, reusing the terms to build the result."
244 `(macrolet ((lc (x) `(term-coeff (car ,x))))
245 (do ((p ,p)
246 (q ,q)
247 r)
248 ((or (endp p) (endp q))
249 ;; NOTE: R contains the result in reverse order. Can it
250 ;; be more efficient to produce the terms in correct order?
251 (unless (endp q)
252 ;; Upon subtraction, we must change the sign of
253 ;; all coefficients in q
254 ,@(when uminus-fn
255 `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))
256 (setf r (nreconc r q)))
257 (unless (endp p)
258 (setf r (nreconc r p)))
259 r)
260 (multiple-value-bind
261 (greater-p equal-p)
262 (funcall ,order-fn (car p) (car q))
263 (cond
264 (greater-p
265 (rotatef (cdr p) r p)
266 )
267 (equal-p
268 (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))
269 (cond
270 ((universal-zerop s)
271 (setf p (cdr p))
272 )
273 (t
274 (setf (lc p) s)
275 (rotatef (cdr p) r p))))
276 (setf q (cdr q))
277 )
278 (t
279 ;;Negate the term of Q if UMINUS provided, signallig
280 ;;that we are doing subtraction
281 ,(when uminus-fn
282 `(setf (lc q) (funcall ,uminus-fn (lc q))))
283 (rotatef (cdr q) r q))))
284 ;;(format t "P:~A~%" p)
285 ;;(format t "Q:~A~%" q)
286 ;;(format t "R:~A~%" r)
287 )))
288
289
290
291(defgeneric add-to (self other)
292 (:documentation "Add OTHER to SELF.")
293 (:method ((self number) (other number))
294 (+ self other))
295 (:method ((self poly) (other number))
296 (add-to self (make-poly-constant (poly-dimension self) other)))
297 (:method ((self number) (other poly))
298 (add-to (make-poly-constant (poly-dimension other) self) other)))
299
300
301(defgeneric subtract-from (self other)
302 (:documentation "Subtract OTHER from SELF.")
303 (:method ((self number) (other number))
304 (- self other))
305 (:method ((self poly) (other number))
306 (subtract-from self (make-poly-constant (poly-dimension self) other))))
307
308
309#|
310(defmacro def-add/subtract-method (add/subtract-method-name
311 uminus-method-name
312 &optional
313 (doc-string nil doc-string-supplied-p))
314 "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."
315 `(defmethod ,add/subtract-method-name ((self poly) (other poly))
316 ,@(when doc-string-supplied-p `(,doc-string))
317 ;; Ensure orders are compatible
318 (change-term-order other self)
319 (setf (poly-termlist self) (fast-add/subtract
320 (poly-termlist self) (poly-termlist other)
321 (poly-term-order self)
322 #',add/subtract-method-name
323 ,(when uminus-method-name `(function ,uminus-method-name))))
324 self))
325
326(eval-when (:load-toplevel :execute)
327
328 (def-add/subtract-method add-to nil
329 "Adds to polynomial SELF another polynomial OTHER.
330This operation destructively modifies both polynomials.
331The result is stored in SELF. This implementation does
332no consing, entirely reusing the sells of SELF and OTHER.")
333
334 (def-add/subtract-method subtract-from unary-minus
335 "Subtracts from polynomial SELF another polynomial OTHER.
336This operation destructively modifies both polynomials.
337The result is stored in SELF. This implementation does
338no consing, entirely reusing the sells of SELF and OTHER.")
339 )
340
341|#
342
343(defmethod unary-minus ((self poly))
344 "Destructively modifies the coefficients of the polynomial SELF,
345by changing their sign."
346 (mapc #'unary-minus (poly-termlist self))
347 self)
348
349(defun add-termlists (p q order-fn)
350 "Destructively adds two termlists P and Q ordered according to ORDER-FN."
351 (fast-add/subtract p q order-fn #'add-to nil))
352
353(defun subtract-termlists (p q order-fn)
354 "Destructively subtracts two termlists P and Q ordered according to ORDER-FN."
355 (fast-add/subtract p q order-fn #'subtract-from #'unary-minus))
356
357(defmethod add-to ((self poly) (other poly))
358 "Adds to polynomial SELF another polynomial OTHER.
359This operation destructively modifies both polynomials.
360The result is stored in SELF. This implementation does
361no consing, entirely reusing the sells of SELF and OTHER."
362 (change-term-order other self)
363 (setf (poly-termlist self) (add-termlists
364 (poly-termlist self) (poly-termlist other)
365 (poly-term-order self)))
366 self)
367
368
369(defmethod subtract-from ((self poly) (other poly))
370 "Subtracts from polynomial SELF another polynomial OTHER.
371This operation destructively modifies both polynomials.
372The result is stored in SELF. This implementation does
373no consing, entirely reusing the sells of SELF and OTHER."
374 (change-term-order other self)
375 (setf (poly-termlist self) (subtract-termlists
376 (poly-termlist self) (poly-termlist other)
377 (poly-term-order self)))
378 self)
379
380(defmacro multiply-term-by-termlist-dropping-zeros (term termlist
381 &optional (reverse-arg-order-P nil))
382 "Multiplies term TERM by a list of term, TERMLIST.
383Takes into accound divisors of zero in the ring, by
384deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P
385is T, change the order of arguments; this may be important
386if we extend the package to non-commutative rings."
387 `(mapcan #'(lambda (other-term)
388 (let ((prod (multiply
389 ,@(cond
390 (reverse-arg-order-p
391 `(other-term ,term))
392 (t
393 `(,term other-term))))))
394 (cond
395 ((universal-zerop prod) nil)
396 (t (list prod)))))
397 ,termlist))
398
399(defun multiply-termlists (p q order-fn)
400 "A version of polynomial multiplication, operating
401directly on termlists."
402 (cond
403 ((or (endp p) (endp q))
404 ;;p or q is 0 (represented by NIL)
405 nil)
406 ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q
407 ((endp (cdr p))
408 (multiply-term-by-termlist-dropping-zeros (car p) q))
409 ((endp (cdr q))
410 (multiply-term-by-termlist-dropping-zeros (car q) p t))
411 (t
412 (cons (multiply (car p) (car q))
413 (add-termlists
414 (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))
415 (multiply-termlists (cdr p) q order-fn)
416 order-fn)))))
417
418(defmethod multiply-by ((self poly) (other poly))
419 (change-term-order other self)
420 (setf (poly-termlist self) (multiply-termlists (poly-termlist self)
421 (poly-termlist other)
422 (poly-term-order self)))
423 self)
424
425(defgeneric add-2 (object1 object2)
426 (:documentation "Non-destructively add OBJECT1 to OBJECT2.")
427 (:method ((object1 t) (object2 t))
428 (add-to (copy-instance object1) (copy-instance object2))))
429
430(defun add (&rest summands)
431 "Non-destructively adds list SUMMANDS."
432 (cond ((endp summands) 0)
433 (t (reduce #'add-2 summands))))
434
435(defun subtract (minuend &rest subtrahends)
436 "Non-destructively subtract MINUEND and SUBTRAHENDS."
437 (cond ((endp subtrahends) (unary-minus minuend))
438 (t (subtract-from (copy-instance minuend) (reduce #'add subtrahends)))))
439
440(defmethod left-tensor-product-by ((self poly) (other monom))
441 (setf (poly-termlist self)
442 (mapcan #'(lambda (term)
443 (let ((prod (left-tensor-product-by term other)))
444 (cond
445 ((universal-zerop prod) nil)
446 (t (list prod)))))
447 (poly-termlist self)))
448 (incf (poly-dimension self) (monom-dimension other))
449 self)
450
451(defmethod right-tensor-product-by ((self poly) (other monom))
452 (setf (poly-termlist self)
453 (mapcan #'(lambda (term)
454 (let ((prod (right-tensor-product-by term other)))
455 (cond
456 ((universal-zerop prod) nil)
457 (t (list prod)))))
458 (poly-termlist self)))
459 (incf (poly-dimension self) (monom-dimension other))
460 self)
461
462
463(defun standard-extension (plist &aux (k (length plist)) (i 0))
464 "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]
465is a list of polynomials. Destructively modifies PLIST elements."
466 (mapc #'(lambda (poly)
467 (left-tensor-product-by
468 poly
469 (prog1
470 (make-monom-variable k i)
471 (incf i))))
472 plist))
473
474(defun standard-extension-1 (plist
475 &aux
476 (plist (standard-extension plist))
477 (nvars (poly-dimension (car plist))))
478 "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].
479Firstly, new K variables U1, U2, ..., UK, are inserted into each
480polynomial. Subsequently, P1, P2, ..., PK are destructively modified
481tantamount to replacing PI with UI*PI-1. It assumes that all
482polynomials have the same dimension, and only the first polynomial
483is examined to determine this dimension."
484 ;; Implementation note: we use STANDARD-EXTENSION and then subtract
485 ;; 1 from each polynomial; since UI*PI has no constant term,
486 ;; we just need to append the constant term at the end
487 ;; of each termlist.
488 (flet ((subtract-1 (p)
489 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
490 (setf plist (mapc #'subtract-1 plist)))
491 plist)
492
493
494(defun standard-sum (plist
495 &aux
496 (plist (standard-extension plist))
497 (nvars (poly-dimension (car plist))))
498 "Calculate the polynomial U1*P1+U2*P2+...+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, and the resulting polynomials
502are added. Finally, 1 is subtracted. It should be noted that the term
503order is not modified, which is equivalent to using a lexicographic
504order on the first K variables."
505 (flet ((subtract-1 (p)
506 (poly-append-term p (make-instance 'term :dimension nvars :coeff -1))))
507 (subtract-1
508 (make-instance
509 'poly
510 :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))
511
512(defgeneric universal-ezgcd (x y)
513 (:documentation "Solves the diophantine system: X=C*X1, Y=C*X2,
514C=GCD(X,Y). It returns C, X1 and Y1. The result may be obtained by
515the Euclidean algorithm.")
516 (:method ((x integer) (y integer)
517 &aux (c (gcd x y)))
518 (values c (/ x c) (/ y c)))
519 )
520
521(defgeneric s-polynomial (object1 object2)
522 (:documentation "Yields the S-polynomial of OBJECT1 and OBJECT2.")
523 (:method ((f poly) (g poly))
524 (let* ((lcm (universal-lcm (leading-monomial f) (leading-monomial g)))
525 (mf (divide lcm (leading-monomial f)))
526 (mg (divide lcm (leading-monomial g))))
527 (multiple-value-bind (c cf cg)
528 (universal-ezgcd (leading-coefficient f) (leading-coefficient g))
529 (declare (ignore c))
530 (subtract
531 (multiply f (change-class mf 'term :coeff cg))
532 (multiply g (change-class mg 'term :coeff cf)))))))
533
534(defgeneric poly-content (object)
535 (:documentation "Greatest common divisor of the coefficients of the polynomial object OBJECT.")
536 (:method ((self poly))
537 (reduce #'universal-gcd
538 (mapcar #'term-coeff (rest (poly-termlist self)))
539 :initial-value (leading-coefficient self))))
540
541(defun poly-primitive-part (object)
542 "Divide polynomial OBJECT by gcd of its
543coefficients. Return the resulting polynomial."
544 (scalar-divide-by object (poly-content object)))
545
546(defun poly-insert-variables (self k)
547 (left-tensor-product-by self (make-instance 'monom :dimension k)))
548
549(defun saturation-extension (f plist &aux (k (length plist)))
550 "Calculate [F', U1*P1-1,U2*P2-1,...,UK*PK-1], where
551PLIST=[P1,P2,...,PK] and F' is F with variables U1,U2,...,UK inserted
552as first K variables. It destructively modifies F and PLIST."
553 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
554 (standard-extension-1 plist)))
555
556(defun polysaturation-extension (f plist &aux (k (length plist)))
557 "Calculate [F', U1*P1+U2*P2+...+UK*PK-1], where PLIST=[P1,P2,...,PK]
558and F' is F with variables U1,U2,...,UK inserted as first K
559variables. It destructively modifies F and PLIST."
560 (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)
561 (list (standard-sum plist))))
562
563(defun saturation-extension-1 (f p)
564 "Given family of polynomials F and a polynomial P, calculate [F',
565U*P-1], where F' is F with variable inserted as the first variable. It
566destructively modifies F and P."
567 (polysaturation-extension f (list p)))
568
569(defmethod multiply-by ((object1 number) (object2 poly))
570 (scalar-multiply-by (copy-instance object2) object1))
571
572(defun make-poly-variable (nvars pos &optional (power 1))
573 (change-class (make-monom-variable nvars pos power) 'poly))
574
575(defun make-poly-constant (nvars coeff)
576 (change-class (make-term-constant nvars coeff) 'poly))
577
578(defgeneric universal-expt (x y)
579 (:documentation "Raises X to power Y.")
580 (:method ((x number) (y integer)) (expt x y))
581 (:method ((x t) (y integer))
582 (declare (type fixnum y))
583 (cond
584 ((minusp y) (error "universal-expt: Negative exponent."))
585 ((universal-zerop x) (if (zerop y) 1))
586 (t
587 (do ((k 1 (ash k 1))
588 (q x (multiply q q)) ;keep squaring
589 (p 1 (if (not (zerop (logand k y))) (multiply p q) p)))
590 ((> k y) p)
591 (declare (fixnum k)))))))
592
593(defgeneric poly-p (object)
594 (:documentation "Checks if an object is a polynomial.")
595 (:method ((self poly)) t)
596 (:method ((self t)) nil))
597
598(defmethod ->sexp :before ((self poly) &optional vars)
599 "Ensures that the number of variables in VARS maches the polynomial dimension of the
600polynomial SELF."
601 (with-slots (dimension)
602 self
603 (assert (= (length vars) dimension)
604 nil
605 "Number of variables ~S does not match the dimension ~S"
606 vars dimension)))
607
608(defmethod ->sexp ((self poly) &optional vars)
609 "Converts a polynomial SELF to a sexp."
610 (let ((m (mapcar #'(lambda (x) (->sexp x vars))
611 (poly-termlist self))))
612 (cond ((endp m) 0)
613 ((endp (cdr m)) (car m))
614 (t (cons '+ m)))))
615
616(defparameter +list-marker+ :[
617 "A sexp with this head is considered a list of polynomials.")
618
619(defmethod ->sexp ((self cons) &optional vars)
620 (assert (eql (car self) +list-marker+))
621 (cons +list-marker+ (mapcar #'(lambda (p) (->sexp p vars)) (cdr self))))
622
623
624(defun poly-eval (expr vars order)
625 "Evaluate Lisp form EXPR to a polynomial or a list of polynomials in
626variables VARS. Return the resulting polynomial or list of
627polynomials. Standard arithmetical operators in form EXPR are
628replaced with their analogues in the ring of polynomials, and the
629resulting expression is evaluated, resulting in a polynomial or a list
630of polynomials in internal form. A similar operation in another computer
631algebra system could be called 'expand' or so."
632 (labels ((p-eval (p) (poly-eval p vars order))
633 (p-eval-list (plist) (mapcar #'p-eval plist)))
634 (cond
635 ((eq expr 0)
636 (make-instance 'poly :dimension (length vars)))
637 ((member expr vars :test #'equalp)
638 (let ((pos (position expr vars :test #'equalp)))
639 (make-poly-variable (length vars) pos)))
640 ((atom expr)
641 (make-poly-constant (length vars) expr))
642 ((eq (car expr) +list-marker+)
643 (cons +list-marker+ (p-eval-list (cdr expr))))
644 (t
645 (case (car expr)
646 (+ (reduce #'add (p-eval-list (cdr expr))))
647 (- (apply #'subtract (p-eval-list (cdr expr))))
648 (*
649 (if (endp (cddr expr)) ;unary
650 (p-eval (cadr expr))
651 (reduce #'multiply (p-eval-list (cdr expr)))))
652 (/
653 ;; A polynomial can be divided by a scalar
654 (cond
655 ((endp (cddr expr))
656 ;; A special case (/ ?), the inverse
657 (divide (cadr expr)))
658 (t
659 (let ((num (p-eval (cadr expr)))
660 (denom-inverse (apply #'divide (mapcar #'p-eval (cddr expr)))))
661 (multiply denom-inverse num)))))
662 (expt
663 (cond
664 ((member (cadr expr) vars :test #'equalp)
665 ;;Special handling of (expt var pow)
666 (let ((pos (position (cadr expr) vars :test #'equalp)))
667 (make-poly-variable (length vars) pos (caddr expr))))
668 ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
669 ;; Negative power means division in coefficient ring
670 ;; Non-integer power means non-polynomial coefficient
671 expr)
672 (t (universal-expt (p-eval (cadr expr)) (caddr expr)))))
673 (otherwise
674 (error "Cannot evaluate as polynomial: ~A" expr)))))))
675
676(defgeneric make-zero-for (self)
677 (:method ((self poly))
678 (make-instance 'poly :dimension (poly-dimension self))))
679
680(defgeneric make-unit-for (self)
681 (:method ((self poly))
682 (make-poly-constant (poly-dimension self) 1)))
683
684(defgeneric poly-nreverse (self)
685 (:documentation "Reverse the order of terms in a polynomial SELF.")
686 (:method ((self poly))
687 (with-slots (termlist)
688 self
689 (setf termlist (nreverse termlist)))
690 self))
691
692
693
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