source: branches/f4grobner/polynomial.lisp@ 3786

;;----------------------------------------------------------------

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

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(defpackage "POLYNOMIAL"

  (:use :cl :utils :monom :copy)

  (:export "POLY"

           "POLY-DIMENSION"

           "POLY-TERMLIST"

           "POLY-TERM-ORDER"

           "POLY-INSERT-TERM"

           "SCALAR-MULTIPLY-BY"

           "SCALAR-DIVIDE-BY"

           "LEADING-TERM"

           "LEADING-MONOMIAL"

           "LEADING-COEFFICIENT"

           "SECOND-LEADING-TERM"

           "SECOND-LEADING-MONOMIAL"

           "SECOND-LEADING-COEFFICIENT"

           "ADD-TO"

           "ADD"

           "SUBTRACT-FROM"

           "SUBTRACT"

           "CHANGE-TERM-ORDER"

           "STANDARD-EXTENSION"

           "STANDARD-EXTENSION-1"

           "STANDARD-SUM"

           "SATURATION-EXTENSION"

           "ALIST->POLY"

           "UNIVERSAL-EZGCD"

           "S-POLYNOMIAL"

           "POLY-CONTENT"

           "POLY-PRIMITIVE-PART"

           "SATURATION-EXTENSION-1"

           "MAKE-POLY-VARIABLE"

           "UNIVERSAL-EXPT"

           "POLY-P")

  (:documentation "Implements polynomials. A polynomial is essentially

a mapping of monomials of the same degree to coefficients. The

momomials are ordered according to a monomial order."))



(in-package :polynomial)



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



(defclass poly ()

  ((dimension :initform nil 

              :initarg :dimension 

              :accessor poly-dimension

              :documentation "Shared dimension of all terms, the number of variables")

   (termlist :initform nil :initarg :termlist :accessor poly-termlist

             :documentation "List of terms.")

   (order :initform #'lex> :initarg :order :accessor poly-term-order

          :documentation "Monomial/term order."))

  (:default-initargs :dimension nil :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)

  (print-unreadable-object (self stream :type t :identity t)

    (with-accessors ((dimension poly-dimension)

                     (termlist poly-termlist)

                     (order poly-term-order))

        self

      (format stream "DIMENSION=~A TERMLIST=~A ORDER=~A" 

              dimension termlist order))))



(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))



(defgeneric poly-insert-term (self term)

  (:documentation "Insert a term TERM into SELF before all other

  terms. Order is not enforced.")

  (:method ((self poly) (term term))

    (cond ((null (poly-dimension self))

           (setf (poly-dimension self) (monom-dimension term)))

          (t (assert (= (poly-dimension self) (monom-dimension term)))))

    (push term (poly-termlist self))

    self))



(defgeneric poly-append-term (self term)

  (:documentation "Append a term TERM to SELF after all other terms. Order is not enforced.")

  (:method ((self poly) (term term))

    (cond ((null (poly-dimension self))

           (setf (poly-dimension self) (monom-dimension term)))

          (t (assert (= (poly-dimension self) (monom-dimension term)))))

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

    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)

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



(defmethod update-instance-for-different-class :after ((old term) (new poly) &key)

  "Converts OLD of class TERM to a NEW of class POLY, by making it into a 1-element TERMLIST."

  (reinitialize-instance new

                         :dimension (monom-dimension old)

                         :termlist (list old)))

  

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

  "Implements equality of polynomials."

  (and (eql (poly-dimension self) (poly-dimension other))

       (every #'universal-equalp (poly-termlist self) (poly-termlist other))

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



(defgeneric leading-term (object)

  (:method ((self poly)) 

    (car (poly-termlist self)))

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



(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."))



(defgeneric leading-monomial (object)

  (:method ((self poly)) 

    (change-class (copy-instance (leading-term self)) 'monom))

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



(defgeneric second-leading-monomial (object)

  (:method ((self poly)) 

    (change-class (copy-instance (second-leading-term self)) 'monom))

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



(defgeneric leading-coefficient (object)

  (:method ((self poly))

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

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



(defgeneric second-leading-coefficient (object)

  (:method ((self poly)) 

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

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

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



(defmethod universal-zerop ((self poly))

  "Return T iff SELF is a zero polynomial."

  (null (poly-termlist self)))



(defgeneric poly-length (self)

  (:documentation "Return the number of terms.")

  (:method ((self poly))

    (length (poly-termlist self))))



(defgeneric scalar-multiply-by (self other)

  (:documentation "Multiply vector SELF by a scalar OTHER.")

  (:method ((self poly) other)

    (mapc #'(lambda (term) (setf (term-coeff term) (multiply (term-coeff term) other)))

          (poly-termlist self))

    self))



(defgeneric scalar-divide-by (self other)

  (:documentation "Divide vector SELF by a scalar OTHER.")

  (:method ((self poly) other)

    (mapc #'(lambda (term) (setf (term-coeff term) (divide (term-coeff term) other)))

          (poly-termlist self))

    self))



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

  "Multiply a polynomial SELF by OTHER."

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

        (poly-termlist self))

  self)



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

  "Multiply a polynomial SELF by OTHER."

  (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) `(term-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 

                ((universal-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)))))))





(defgeneric add-to (self other)

  (:documentation "Add OTHER to SELF.")

  (:method ((self number) (other number))

    (+ self other)))



(defgeneric subtract-from (self other)

  (:documentation "Subtract OTHER from SELF.")

  (:method ((self number) (other number))

    (- self other)))



(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 (multiply

                            ,@(cond 

                               (reverse-arg-order-p

                                `(other-term ,term))

                               (t 

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

                 (cond 

                   ((universal-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 (multiply (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)



(defun add (object1 object2)

  "Non-destructively add POLY1 by POLY2."

  (add-to (copy-instance object1) (copy-instance object2)))



(defun subtract (minuend  &rest subtrahends)

  "Non-destructively subtract MINUEND and SUBTRAHENDS."

  (subtract-from (copy-instance minuend) (reduce #'add subtrahends)))



(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 

                        ((universal-zerop prod) nil)

                        (t (list prod)))))

                (poly-termlist self)))

  (incf (poly-dimension self) (monom-dimension other))

  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 

                        ((universal-zerop prod) nil)

                        (t (list prod)))))

                (poly-termlist self)))

  (incf (poly-dimension self) (monom-dimension other))

  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))



(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) 

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

    (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) 

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

    (subtract-1 

     (make-instance

      'poly

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



(defgeneric universal-ezgcd (x y)

  (:documentation "Solves the diophantine system: X=C*X1, Y=C*X2,

C=GCD(X,Y).  It returns C, X1 and Y1. The result may be obtained by

the Euclidean algorithm.")

  (:method ((x integer) (y integer)

            &aux (c (gcd x y)))

    (values c (/ x c) (/ y c)))

  )



(defgeneric s-polynomial (object1 object2)

  (:documentation "Yields the S-polynomial of OBJECT1 and OBJECT2.")

  (:method ((f poly) (g poly))

    (let* ((lcm (universal-lcm (leading-monomial f) (leading-monomial g)))

           (mf (divide lcm (leading-monomial f)))

           (mg (divide lcm (leading-monomial g))))

      (multiple-value-bind (c cf cg)

          (universal-ezgcd (leading-coefficient f) (leading-coefficient g))

        (declare (ignore c))

        (subtract

         (multiply f (change-class mf 'term :coeff cg))

         (multiply g (change-class mg 'term :coeff cf)))))))



(defgeneric poly-content (object)

  (:documentation "Greatest common divisor of the coefficients of the polynomial object OBJECT.")

  (:method ((self poly))

    (reduce #'universal-gcd 

            (mapcar #'term-coeff (rest (poly-termlist self))) 

            :initial-value (leading-coefficient self))))



(defun poly-primitive-part (object)

  "Divide polynomial OBJECT by gcd of its

coefficients. Return the resulting polynomial."

  (scalar-divide-by object (poly-content object)))



(defun poly-insert-variables (self k)

  (left-tensor-product-by self (make-instance 'monom :dimension k)))



(defun saturation-extension (f plist &aux (k (length plist)))

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

PLIST=[P1,P2,...,PK] and F' is F with variables U1,U2,...,UK inserted

as first K variables. It destructively modifies F and PLIST."

  (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)

         (standard-extension-1 plist)))



(defun polysaturation-extension (f plist &aux (k (length plist)))

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

and F' is F with variables U1,U2,...,UK inserted as first K

variables. It destructively modifies F and PLIST."

  (nconc (mapc #'(lambda (x) (poly-insert-variables x k)) f)

         (list (standard-sum plist))))



(defun saturation-extension-1 (f p) 

  "Given family of polynomials F and a polynomial P, calculate [F',

U*P-1], where F' is F with variable inserted as the first variable. It

destructively modifies F and P."

  (polysaturation-extension f (list p)))



(defmethod multiply-by ((object1 number) (object2 poly))

  (scalar-multiply-by (copy-instance object2) object1))



(defun make-poly-variable (nvars pos &optional (power 1))

  (change-class (make-monom-variable nvars pos power) 'poly))



(defgeneric universal-expt (x y)

  (:documentation "Raises X to power Y.")

  (:method ((x number) (y integer)) (expt x y))

  (:method ((x t) (y integer)) 

    (declare (type fixnum y))

    (cond

      ((minusp y) (error "universal-expt: Negative exponent."))

      ((universal-zerop x) (if (zerop y) 1))

      (t

       (do ((k 1 (ash k 1))

            (q x (multiply q q))        ;keep squaring

            (p 1 (if (not (zerop (logand k y))) (multiply p q) p)))

           ((> k y) p)

         (declare (fixnum k)))))))



(defgeneric poly-p (object)

  (:documentation "Checks if an object is a polynomial.")

  (:method ((self poly)) t)

  (:method ((self t)) nil))