source: branches/f4grobner/monom.lisp@ 1904

;;; -*-  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.                                

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

;; 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)

  (:export "MONOM"

           "EXPONENT"

           "MAKE-MONOM" 

           "MAKE-MONOM-VARIABLE" 

           "MONOM-ELT" 

           "MONOM-DIMENSION" 

           "MONOM-TOTAL-DEGREE"

           "MONOM-SUGAR" 

           "MONOM-DIV"

           "MONOM-MUL" 

           "MONOM-DIVIDES-P"

           "MONOM-DIVIDES-MONOM-LCM-P"

           "MONOM-LCM-DIVIDES-MONOM-LCM-P"

           "MONOM-LCM-EQUAL-MONOM-LCM-P"

           "MONOM-DIVISIBLE-BY-P"

           "MONOM-REL-PRIME-P"

           "MONOM-EQUAL-P"

           "MONOM-LCM"

           "MONOM-GCD"

           "MONOM-DEPENDS-P"

           "MONOM-MAP"

           "MONOM-APPEND"

           "MONOM-CONTRACT"

           "MONOM->LIST"))



(in-package :monom)



(deftype exponent ()

  "Type of exponent in a monomial."

  'fixnum)



(deftype monom (&optional dim)

  "Type of monomial."

  `(simple-array exponent (,dim)))



;; 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 nor INITIAL-EXPONENTS"))))

                     (monom (cond 

                              ;; when exponents are supplied

                              (initial-exponents-supplied-p

                               (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."

  monom)



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

;;

;; Operations on monomials

;;

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



(defun monom-dimension (m)

  (declare (type monom m))

  (length m))



(defmacro monom-elt (m index)

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

  `(elt ,m ,index))



(defun monom-total-degree (m &optional (start 0) (end (monom-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 monom m) (type fixnum start end))

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



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

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

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

  (declare (type monom m) (type fixnum start end))

  (monom-total-degree m start end))



(defun monom-div (m1 m2 &aux (result (copy-seq m1)))

  "Divide monomial M1 by monomial M2."

  (declare (type monom m1 m2 result))

  (map-into result #'- m1 m2))



(defun monom-mul (m1 m2  &aux (result (copy-seq m1)))

  "Multiply monomial M1 by monomial M2."

  (declare (type monom m1 m2 result))

  (map-into result #'+ m1 m2))



(defun monom-divides-p (m1 m2)

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

  (declare (type monom m1 m2))

  (every #'<= m1 m2))



(defun monom-divides-monom-lcm-p (m1 m2 m3)

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

  (declare (type monom m1 m2 m3))

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

         m1 m2 m3))



(defun monom-lcm-divides-monom-lcm-p (m1 m2 m3 m4)

  "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))

         



(defun monom-lcm-equal-monom-lcm-p (m1 m2 m3 m4)

  "Returns T if monomial MONOM-LCM(M1,M2) equals 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))





(defun monom-divisible-by-p (m1 m2)

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

  (declare (type monom m1 m2))

  (every #'>= m1 m2))



(defun monom-rel-prime-p (m1 m2)

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

  (declare (type monom m1 m2))

  (every #'(lambda (x y) (zerop (min x y))) m1 m2))



(defun monom-equal-p (m1 m2)

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

  (declare (type monom m1 m2))

  (every #'= m1 m2))



(defun monom-lcm (m1 m2 &aux (result (copy-seq m1)))

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

  (declare (type monom m1 m2 result))

  (map-into result #'max m1 m2))



(defun monom-gcd (m1 m2 &aux (result (copy-seq m1)))

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

  (declare (type monom m1 m2 result))

  (map-into result #'min m1 m2))



(defun monom-depends-p (m k)

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

  (declare (type monom m) (type fixnum k))

  (plusp (monom-elt m k)))



(defmacro monom-map (fun m &rest ml &aux (result `(copy-seq ,m)))

  "Map function FUN of one argument over the powers of a monomial M.

Fun should map a single FIXNUM argument to FIXNUM.  Return a sequence

of results."

  `(map-into ,result ,fun ,m ,@ml))



(defun monom-append (m1 m2 &aux (dim (+ (length m1) (length m2))))

  (declare (type monom m1 m2) (fixnum dim))

  (concatenate `(monom ,dim) m1 m2))



(defmacro monom-contract (m k)

  "Drop the first K variables in monomial M."

  `(setf ,m (subseq ,m ,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))

  (setf (monom-elt m pos) power)

  m)



(defun monom->list (m)

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

  (declare (type monom m))

  (coerce m 'list))