source: branches/f4grobner/monom.lisp@ 2010

;;; -*-  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)

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

(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))

(defun monom-contract (m k)
  "Drop the first K variables in monomial M."
  (declare (type monom m) (fixnum k))
  (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) (type monom m))
  (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))