source: branches/f4grobner/monom.lisp@ 2191

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

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;;;                                                                              

;;;  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.                                

;;;                                                                              

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

;; This package implements BASIC OPERATIONS ON MONOMIALS

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

;; DATA STRUCTURES: Conceptually, monomials can be represented as lists:

;;

;;      monom:  (n1 n2 ... nk) where ni are non-negative integers

;;

;; However, lists may be implemented as other sequence types,

;; so the flexibility to change the representation should be

;; maintained in the code to use general operations on sequences

;; whenever possible. The optimization for the actual representation

;; should be left to declarations and the compiler.

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

;; EXAMPLES: Suppose that variables are x and y. Then

;;

;;      Monom x*y^2 ---> (1 2)

;;

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



(defpackage "MONOM"

  (:use :cl :ring)

  (:export "MONOM"

           "EXPONENT"

           "MAKE-MONOM"

           "MONOM-DIMENSION"

           "MONOM-EXPONENTS"

           "MAKE-MONOM-VARIABLE"))



(in-package :monom)



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



(deftype exponent ()

  "Type of exponent in a monomial."

  'fixnum)



(defclass monom ()

  ((dim       :initarg :dim :accessor monom-dimension)

   (exponents :initarg :exponents :accessor monom-exponents))

  (:default-initargs :dim 0 :exponents nil))



(defmethod print-object ((m monom) stream)

  (princ (slot-value m 'exponents) stream))



;; If a monomial is redefined as structure with slot EXPONENTS, the function

;; below can be the BOA constructor.

(defun make-monom (&key 

                     (dimension nil dimension-suppied-p)

                     (initial-exponents nil initial-exponents-supplied-p)

                     (initial-exponent  nil initial-exponent-supplied-p)

                   &aux

                     (dim (cond (dimension-suppied-p dimension)

                                (initial-exponents-supplied-p (length initial-exponents))

                                (t (error "You must provide DIMENSION or INITIAL-EXPONENTS"))))

                     (exponents (cond 

                                  ;; when exponents are supplied

                                  (initial-exponents-supplied-p

                                   (when (and dimension-suppied-p (/= dimension (length initial-exponents)))

                                     (error "INITIAL-EXPONENTS must have length DIMENSION"))

                                   (make-array (list dim) :initial-contents initial-exponents

                                               :element-type 'exponent))

                                  ;; when all exponents are to be identical

                                  (initial-exponent-supplied-p

                                   (make-array (list dim) :initial-element initial-exponent

                                               :element-type 'exponent))

                                  ;; otherwise, all exponents are zero

                                  (t 

                                   (make-array (list dim) :element-type 'exponent :initial-element 0)))))

  "A constructor (factory) of monomials. If DIMENSION is given, a sequence of

DIMENSION elements of type EXPONENT is constructed, where individual

elements are the value of INITIAL-EXPONENT, which defaults to 0.

Alternatively, all elements may be specified as a list

INITIAL-EXPONENTS."

  (make-instance 'monom :dim dim :exponents exponents))



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

;;

;; Operations on monomials

;;

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



(defmethod r-dimension ((m monom))

  (monom-dimension m))



(defmethod r-elt ((m monom) index)

  "Return the power in the monomial M of variable number INDEX."

  (with-slots (exponents)

      m

    (elt exponents index)))



(defmethod (setf r-elt) (new-value (m monom) index)

  "Return the power in the monomial M of variable number INDEX."

  (with-slots (exponents)

      m

    (setf (elt exponents index) new-value)))



(defmethod r-total-degree ((m monom) &optional (start 0) (end (r-dimension m)))

  "Return the todal degree of a monomoal M. Optinally, a range

of variables may be specified with arguments START and END."

  (declare (type fixnum start end))

  (with-slots (exponents)

      m

    (reduce #'+ exponents :start start :end end)))





(defmethod r-sugar ((m monom) &aux (start 0) (end (r-dimension m)))

  "Return the sugar of a monomial M. Optinally, a range

of variables may be specified with arguments START and END."

  (declare (type fixnum start end))

    (r-total-degree m start end))



(defmethod r* ((m1 monom) (m2 monom))

  "Multiply monomial M1 by monomial M2."

  (with-slots ((exponents1 exponents) dim)

      m1

    (with-slots ((exponents2 exponents))

        m2

      (let* ((exponents (copy-seq exponents1)))

        (map-into exponents #'+ exponents1 exponents2)

        (make-instance 'monom :dim dim :exponents exponents)))))







(defmethod r/ ((m1 monom) (m2 monom))

  "Divide monomial M1 by monomial M2."

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (let* ((exponents (copy-seq exponents1))

             (dim (reduce #'+ exponents)))

        (map-into exponents #'- exponents1 exponents2)

        (make-instance 'monom :dim dim :exponents exponents)))))



(defmethod r-divides-p ((m1 monom) (m2 monom))

  "Returns T if monomial M1 divides monomial M2, NIL otherwise."

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (every #'<= exponents1 exponents2))))





(defmethod r-divides-lcm-p ((m1 monom) (m2 monom) (m3 monom))

  "Returns T if monomial M1 divides LCM(M2,M3), NIL otherwise."

  (every #'(lambda (x y z) (<= x (max y z))) 

         m1 m2 m3))





(defmethod r-lcm-divides-lcm-p ((m1 monom) (m2 monom) (m3 monom) (m4 monom))

  "Returns T if monomial MONOM-LCM(M1,M2) divides MONOM-LCM(M3,M4), NIL otherwise."

  (declare (type monom m1 m2 m3 m4))

  (every #'(lambda (x y z w) (<= (max x y) (max z w))) 

         m1 m2 m3 m4))

         

(defmethod r-lcm-equal-lcm-p (m1 m2 m3 m4)

  "Returns T if monomial LCM(M1,M2) equals LCM(M3,M4), NIL otherwise."

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (with-slots ((exponents3 exponents))

          m3

        (with-slots ((exponents4 exponents))

            m4

          (every 

           #'(lambda (x y z w) (= (max x y) (max z w)))

           exponents1 exponents2 exponents3 exponents4))))))



(defmethod r-divisible-by-p ((m1 monom) (m2 monom))

  "Returns T if monomial M1 is divisible by monomial M2, NIL otherwise."

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (every #'>= exponents1 exponents2))))



(defmethod r-rel-prime-p ((m1 monom) (m2 monom))

  "Returns T if two monomials M1 and M2 are relatively prime (disjoint)."

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (every #'(lambda (x y) (zerop (min x y))) exponents1 exponents2))))





(defmethod r-equalp ((m1 monom) (m2 monom))

  "Returns T if two monomials M1 and M2 are equal."

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (every #'= exponents1 exponents2))))



(defmethod r-lcm ((m1 monom) (m2 monom)) 

  "Returns least common multiple of monomials M1 and M2."

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (let* ((exponents (copy-seq exponents1))

             (dim (reduce #'+ exponents)))

        (map-into exponents #'max exponents1 exponents2)

        (make-instance 'monom :dim dim :exponents exponents)))))





(defmethod r-gcd ((m1 monom) (m2 monom))

  "Returns greatest common divisor of monomials M1 and M2."

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (let* ((exponents (copy-seq exponents1))

             (dim (reduce #'+ exponents)))

        (map-into exponents #'min exponents1 exponents2)

        (make-instance 'monom :dim dim :exponents exponents)))))



(defmethod r-depends-p ((m monom) k)

  "Return T if the monomial M depends on variable number K."

  (declare (type fixnum k))

  (with-slots (exponents)

      m

    (plusp (elt exponents k))))



(defmethod r-tensor-product ((m1 monom) (m2 monom)

                             &aux (dim (+ (r-dimension m1) (r-dimension m2))))

  (declare (fixnum dim))

  (with-slots ((exponents1 exponents))

      m1

    (with-slots ((exponents2 exponents))

        m2

      (make-instance 'monom 

                     :dim dim

                     :exponents (concatenate 'vector exponents1 exponents2)))))



(defmethod r-contract ((m monom) k)

  "Drop the first K variables in monomial M."

  (declare (fixnum k))

  (with-slots (dim exponents) 

      m

    (setf dim (- dim k)

          exponents (subseq exponents k))))



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

                            &aux (m (make-monom :dimension nvars)))

  "Construct a monomial in the polynomial ring

RING[X[0],X[1],X[2],...X[NVARS-1]] over the (unspecified) ring RING

which represents a single variable. It assumes number of variables

NVARS and the variable is at position POS. Optionally, the variable

may appear raised to power POWER. "

  (declare (type fixnum nvars pos power) (type monom m))

  (with-slots (exponents)

      m

    (setf (elt exponents pos) power)

    m))



(defmethod r->list ((m monom))

  "A human-readable representation of a monomial M as a list of exponents."  

  (coerce (monom-exponents m) 'list))