source: branches/f4grobner/polynomial.lisp@ 3163

;;; -*-  Mode: Lisp -*- 

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;;;                                                                              

;;;  Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>          

;;;                                                                              

;;;  This program is free software; you can redistribute it and/or modify        

;;;  it under the terms of the GNU General Public License as published by        

;;;  the Free Software Foundation; either version 2 of the License, or           

;;;  (at your option) any later version.                                         

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;;;  This program is distributed in the hope that it will be useful,             

;;;  but WITHOUT ANY WARRANTY; without even the implied warranty of              

;;;  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the               

;;;  GNU General Public License for more details.                                

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;;;  You should have received a copy of the GNU General Public License           

;;;  along with this program; if not, write to the Free Software                 

;;;  Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.  

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(defpackage "POLYNOMIAL"

  (:use :cl :utils :ring :monom :order :term)

  (:export "POLY"

           "POLY-TERMLIST"

           "POLY-TERM-ORDER"

           "CHANGE-TERM-ORDER"

           "STANDARD-EXTENSION"

           "STANDARD-EXTENSION-1"

           "STANDARD-SUM"

           "SATURATION-EXTENSION"

           "ALIST->POLY")

  (:documentation "Implements polynomials."))



(in-package :polynomial)



(proclaim '(optimize (speed 3) (space 0) (safety 0) (debug 0)))



(defclass poly ()

  ((termlist :initarg :termlist :accessor poly-termlist

             :documentation "List of terms.")

   (order :initarg :order :accessor poly-term-order

          :documentation "Monomial/term order."))

  (:default-initargs :termlist nil :order #'lex>)

  (:documentation "A polynomial with a list of terms TERMLIST, ordered

according to term order ORDER, which defaults to LEX>."))



(defmethod print-object ((self poly) stream)

  (format stream "#<POLY TERMLIST=~A ORDER=~A>" 

          (poly-termlist self)

          (poly-term-order self)))



(defgeneric change-term-order (self other)

  (:documentation "Change term order of SELF to the term order of OTHER.")

  (:method ((self poly) (other poly))

    (unless (eq (poly-term-order self) (poly-term-order other))

      (setf (poly-termlist self) (sort (poly-termlist self) (poly-term-order other))

            (poly-term-order self) (poly-term-order other)))

    self))



(defun alist->poly (alist &aux (poly (make-instance 'poly)))

  "It reads polynomial from an alist formatted as ( ... (exponents . coeff) ...).

It can be used to enter simple polynomials by hand, e.g the polynomial

in two variables, X and Y, given in standard notation as:



      3*X^2*Y^3+2*Y+7



can be entered as

(ALIST->POLY '(((2 3) . 3) ((0 1) . 2) ((0 0) . 7))).



NOTE: The primary use is for low-level debugging of the package."

  (dolist (x alist poly)

    (insert-item poly (make-instance 'term :exponents (car x) :coeff (cdr x)))))





(defmethod r-equalp ((self poly) (other poly))

  "POLY instances are R-EQUALP if they have the same

order and if all terms are R-EQUALP."

  (and (every #'r-equalp (poly-termlist self) (poly-termlist other))

       (eq (poly-term-order self) (poly-term-order other))))



(defmethod insert-item ((self poly) (item term))

  (push item (poly-termlist self))

  self)



(defmethod append-item ((self poly) (item term))

  (setf (cdr (last (poly-termlist self))) (list item))

  self)



;; Leading term

(defgeneric leading-term (object)

  (:method ((self poly)) 

    (car (poly-termlist self)))

  (:documentation "The leading term of a polynomial, or NIL for zero polynomial."))



;; Second term

(defgeneric second-leading-term (object) 

  (:method ((self poly))

    (cadar (poly-termlist self)))

  (:documentation "The second leading term of a polynomial, or NIL for a polynomial with at most one term."))



;; Leading coefficient

(defgeneric leading-coefficient (object)

  (:method ((self poly))

    (r-coeff (leading-term self)))

  (:documentation "The leading coefficient of a polynomial. It signals error for a zero polynomial."))



;; Second coefficient

(defgeneric second-leading-coefficient (object)

  (:method ((self poly)) 

    (r-coeff (second-leading-term self)))

  (:documentation "The second leading coefficient of a polynomial. It

  signals error for a polynomial with at most one term."))



;; Testing for a zero polynomial

(defmethod r-zerop ((self poly))

  (null (poly-termlist self)))



;; The number of terms

(defmethod r-length ((self poly))

  (length (poly-termlist self)))



(defmethod multiply-by ((self poly) (other monom))

  (mapc #'(lambda (term) (multiply-by term other))

        (poly-termlist self))

  self)



(defmethod multiply-by ((self poly) (other term))

  (mapc #'(lambda (term) (multiply-by term other))

        (poly-termlist self))

  self)



(defmethod multiply-by ((self poly) (other scalar))

  (mapc #'(lambda (term) (multiply-by term other))

        (poly-termlist self))

  self)





(defmacro fast-add/subtract (p q order-fn add/subtract-fn uminus-fn)

  "Return an expression which will efficiently adds/subtracts two

polynomials, P and Q.  The addition/subtraction of coefficients is

performed by calling ADD/SUBTRACT-METHOD-NAME.  If UMINUS-METHOD-NAME

is supplied, it is used to negate the coefficients of Q which do not

have a corresponding coefficient in P. The code implements an

efficient algorithm to add two polynomials represented as sorted lists

of terms. The code destroys both arguments, reusing the terms to build

the result."

  `(macrolet ((lc (x) `(r-coeff (car ,x))))

     (do ((p ,p)

          (q ,q)

          r)

         ((or (endp p) (endp q))

          ;; NOTE: R contains the result in reverse order. Can it

          ;; be more efficient to produce the terms in correct order?

          (unless (endp q) 

            ;; Upon subtraction, we must change the sign of

            ;; all coefficients in q

            ,@(when uminus-fn

                    `((mapc #'(lambda (x) (setf x (funcall ,uminus-fn x))) q)))

            (setf r (nreconc r q)))

          r)

       (multiple-value-bind 

             (greater-p equal-p)

           (funcall ,order-fn (car p) (car q))

         (cond

           (greater-p

            (rotatef (cdr p) r p)

            )

           (equal-p

            (let ((s (funcall ,add/subtract-fn (lc p) (lc q))))

              (cond 

                ((r-zerop s)

                 (setf p (cdr p))

                 )

                (t 

                 (setf (lc p) s)

                 (rotatef (cdr p) r p))))

            (setf q (cdr q))

            )

           (t 

            ;;Negate the term of Q if UMINUS provided, signallig

            ;;that we are doing subtraction

            ,(when uminus-fn

                   `(setf (lc q) (funcall ,uminus-fn (lc q))))

            (rotatef (cdr q) r q)))))))





(defmacro def-add/subtract-method (add/subtract-method-name

                                   uminus-method-name

                                   &optional

                                     (doc-string nil doc-string-supplied-p))

  "This macro avoids code duplication for two similar operations: ADD-TO and SUBTRACT-FROM."

  `(defmethod ,add/subtract-method-name ((self poly) (other poly))

     ,@(when doc-string-supplied-p `(,doc-string))

     ;; Ensure orders are compatible

     (change-term-order other self)

     (setf (poly-termlist self) (fast-add/subtract 

                                 (poly-termlist self) (poly-termlist other)

                                 (poly-term-order self)

                                 #',add/subtract-method-name

                                 ,(when uminus-method-name `(function ,uminus-method-name))))

     self))



(eval-when (:compile-toplevel :load-toplevel :execute)



  (def-add/subtract-method add-to nil

    "Adds to polynomial SELF another polynomial OTHER.

This operation destructively modifies both polynomials.

The result is stored in SELF. This implementation does

no consing, entirely reusing the sells of SELF and OTHER.")



  (def-add/subtract-method subtract-from unary-minus

    "Subtracts from polynomial SELF another polynomial OTHER.

This operation destructively modifies both polynomials.

The result is stored in SELF. This implementation does

no consing, entirely reusing the sells of SELF and OTHER.")

  )



(defmethod unary-minus ((self poly))

  "Destructively modifies the coefficients of the polynomial SELF,

by changing their sign."

  (mapc #'unary-minus (poly-termlist self))

  self)



(defun add-termlists (p q order-fn)

  "Destructively adds two termlists P and Q ordered according to ORDER-FN."

  (fast-add/subtract p q order-fn #'add-to nil))



(defmacro multiply-term-by-termlist-dropping-zeros (term termlist 

                                                    &optional (reverse-arg-order-P nil))

  "Multiplies term TERM by a list of term, TERMLIST.

Takes into accound divisors of zero in the ring, by

deleting zero terms. Optionally, if REVERSE-ARG-ORDER-P

is T, change the order of arguments; this may be important

if we extend the package to non-commutative rings."

  `(mapcan #'(lambda (other-term) 

               (let ((prod (r*

                            ,@(cond 

                               (reverse-arg-order-p

                                `(other-term ,term))

                               (t 

                                `(,term other-term))))))

                 (cond 

                   ((r-zerop prod) nil)

                   (t (list prod)))))

           ,termlist))



(defun multiply-termlists (p q order-fn)

  "A version of polynomial multiplication, operating

directly on termlists."

  (cond 

    ((or (endp p) (endp q))         

     ;;p or q is 0 (represented by NIL)

     nil)       

    ;; If p= p0+p1 and q=q0+q1 then p*q=p0*q0+p0*q1+p1*q

    ((endp (cdr p))

     (multiply-term-by-termlist-dropping-zeros (car p) q))

    ((endp (cdr q))

     (multiply-term-by-termlist-dropping-zeros (car q) p t))

    (t

     (cons (r* (car p) (car q))

           (add-termlists 

            (multiply-term-by-termlist-dropping-zeros (car p) (cdr q))

            (multiply-termlists (cdr p) q order-fn)

            order-fn)))))



(defmethod multiply-by ((self poly) (other poly))

  (change-term-order other self)

  (setf (poly-termlist self) (multiply-termlists (poly-termlist self)

                                                 (poly-termlist other)

                                                 (poly-term-order self)))

  self)



(defmethod r* ((poly1 poly) (poly2 poly))

  "Non-destructively multiply POLY1 by POLY2."

  (multiply-by (copy-instance POLY1) (copy-instance POLY2)))



(defmethod left-tensor-product-by ((self poly) (other term))

  (setf (poly-termlist self) 

        (mapcan #'(lambda (term) 

                    (let ((prod (left-tensor-product-by term other)))

                      (cond 

                        ((r-zerop prod) nil)

                        (t (list prod)))))

                (poly-termlist self)))

  self)



(defmethod right-tensor-product-by ((self poly) (other term))

  (setf (poly-termlist self) 

        (mapcan #'(lambda (term) 

                    (let ((prod (right-tensor-product-by term other)))

                      (cond 

                        ((r-zerop prod) nil)

                        (t (list prod)))))

                (poly-termlist self)))

  self)



(defmethod left-tensor-product-by ((self poly) (other monom))

  (setf (poly-termlist self) 

        (mapcan #'(lambda (term) 

                    (let ((prod (left-tensor-product-by term other)))

                      (cond 

                        ((r-zerop prod) nil)

                        (t (list prod)))))

                (poly-termlist self)))

  self)



(defmethod right-tensor-product-by ((self poly) (other monom))

  (setf (poly-termlist self) 

        (mapcan #'(lambda (term) 

                    (let ((prod (right-tensor-product-by term other)))

                      (cond 

                        ((r-zerop prod) nil)

                        (t (list prod)))))

                (poly-termlist self)))

  self)





(defun standard-extension (plist &aux (k (length plist)) (i 0))

  "Calculate [U1*P1,U2*P2,...,UK*PK], where PLIST=[P1,P2,...,PK]

is a list of polynomials. Destructively modifies PLIST elements."

  (mapc #'(lambda (poly)

            (left-tensor-product-by 

             poly 

             (prog1 

                 (make-monom-variable k i) 

               (incf i))))

        plist))



(defmethod poly-dimension ((poly poly))

  (cond ((r-zerop poly) -1)

        (t (monom-dimension (leading-term poly)))))



(defun standard-extension-1 (plist 

                             &aux 

                               (plist (standard-extension plist))

                               (nvars (poly-dimension (car plist))))

  "Calculate [U1*P1-1,U2*P2-1,...,UK*PK-1], where PLIST=[P1,P2,...,PK].

Firstly, new K variables U1, U2, ..., UK, are inserted into each

polynomial.  Subsequently, P1, P2, ..., PK are destructively modified

tantamount to replacing PI with UI*PI-1. It assumes that all

polynomials have the same dimension, and only the first polynomial

is examined to determine this dimension."

  ;; Implementation note: we use STANDARD-EXTENSION and then subtract

  ;; 1 from each polynomial; since UI*PI has no constant term,

  ;; we just need to append the constant term at the end

  ;; of each termlist.

  (flet ((subtract-1 (p) 

           (append-item p (make-instance 'term :coeff -1 :dimension nvars))))

    (setf plist (mapc #'subtract-1 plist)))

  plist)





(defun standard-sum (plist

                     &aux 

                       (plist (standard-extension plist))

                       (nvars (poly-dimension (car plist))))

  "Calculate the polynomial U1*P1+U2*P2+...+UK*PK-1, where PLIST=[P1,P2,...,PK].

Firstly, new K variables, U1, U2, ..., UK, are inserted into each

polynomial.  Subsequently, P1, P2, ..., PK are destructively modified

tantamount to replacing PI with UI*PI, and the resulting polynomials

are added. Finally, 1 is subtracted.  It should be noted that the term

order is not modified, which is equivalent to using a lexicographic

order on the first K variables."

  (flet ((subtract-1 (p) 

           (append-item p (make-instance 'term :coeff -1 :dimension nvars))))

    (subtract-1 

     (make-instance

      'poly

      :termlist (apply #'nconc (mapcar #'poly-termlist plist))))))



#|



(defun saturation-extension-1 (ring f p) 

  "Calculate [F, U*P-1]. It destructively modifies F."

  (declare (type ring ring))

  (polysaturation-extension ring f (list p)))









(defun spoly (ring-and-order f g

              &aux

                (ring (ro-ring ring-and-order)))

  "It yields the S-polynomial of polynomials F and G."

  (declare (type ring-and-order ring-and-order) (type poly f g))

  (let* ((lcm (monom-lcm (poly-lm f) (poly-lm g)))

         (mf (monom-div lcm (poly-lm f)))

         (mg (monom-div lcm (poly-lm g))))

    (declare (type monom mf mg))

    (multiple-value-bind (c cf cg)

        (funcall (ring-ezgcd ring) (poly-lc f) (poly-lc g))

      (declare (ignore c))

      (poly-sub 

       ring-and-order

       (scalar-times-poly ring cg (monom-times-poly mf f))

       (scalar-times-poly ring cf (monom-times-poly mg g))))))





(defun poly-primitive-part (ring p)

  "Divide polynomial P with integer coefficients by gcd of its

coefficients and return the result."

  (declare (type ring ring) (type poly p))

  (if (poly-zerop p)

      (values p 1)

      (let ((c (poly-content ring p)))

        (values (make-poly-from-termlist 

                 (mapcar

                  #'(lambda (x)

                      (make-term :monom (term-monom x)

                                 :coeff (funcall (ring-div ring) (term-coeff x) c)))

                  (poly-termlist p))

                 (poly-sugar p))

                c))))



(defun poly-content (ring p)

  "Greatest common divisor of the coefficients of the polynomial P. Use the RING structure

to compute the greatest common divisor."

  (declare (type ring ring) (type poly p))

  (reduce (ring-gcd ring) (mapcar #'term-coeff (rest (poly-termlist p))) :initial-value (poly-lc p)))



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