source: branches/f4grobner/monomial.lisp@ 56

;;----------------------------------------------------------------
;; This package implements BASIC OPERATIONS ON MONOMIALS
;;----------------------------------------------------------------
;; DATA STRUCTURES: Monomials are 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)
;;
;;----------------------------------------------------------------

(deftype exponent ()
  "Type of exponent in a monomial."
  'fixnum)

(deftype monom (&optional dim)
  "Type of monomial."
  `(simple-array exponent (,dim)))

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;;
;; Construction of monomials
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defmacro make-monom (dim &key (initial-contents nil initial-contents-supplied-p)
                               (initial-element 0 initial-element-supplied-p))
  "Make a monomial with DIM variables. Additional argument
INITIAL-CONTENTS specifies the list of powers of the consecutive
variables. The alternative additional argument INITIAL-ELEMENT
specifies the common power for all variables."
  ;;(declare (fixnum dim))
  `(make-array ,dim
               :element-type 'exponent
               ,@(when initial-contents-supplied-p `(:initial-contents ,initial-contents))
               ,@(when initial-element-supplied-p `(:initial-element ,initial-element))))



;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Operations on monomials
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defmacro monom-elt (m index)
  "Return the power in the monomial M of variable number INDEX."
  `(elt ,m ,index))

(defun monom-dimension (m)
  "Return the number of variables in the monomial M."
  (length m))

(defun monom-total-degree (m &optional (start 0) (end (length 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) (fixnum start end))
  (reduce #'+ m :start start :end end))

(defun monom-sugar (m &aux (start 0) (end (length 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) (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) (declare (type exponent 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) (declare (type exponent 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) (declare (type exponent 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) (declare (type exponent 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))
  (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))
  (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) (fixnum k))
  (plusp (elt m k)))

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

(defmacro monom-append (m1 m2)
  `(concatenate 'monom ,m1 ,m2))

(defmacro monom-contract (k m)
  `(subseq ,m ,k))

(defun monom-exponents (m)
  (declare (type monom m))
  (coerce m 'list))