path: root/src/algebra/modgcd.spad.pamphlet

\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra modgcd.spad}
\author{James Davenport, Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package INMODGCD InnerModularGcd}
<<package INMODGCD InnerModularGcd>>=
)abbrev package INMODGCD InnerModularGcd
++ Author: J.H. Davenport and P. Gianni
++ Date Created: July 1990
++ Date Last Updated: November 1991
++ Description:
++ This file contains the functions for modular gcd algorithm
++ for univariate polynomials with coefficients in a
++ non-trivial euclidean domain (i.e. not a field).
++ The package parametrised by the coefficient domain,
++ the polynomial domain, a prime,
++ and a function for choosing the next prime

Z    ==> Integer
NNI  ==> NonNegativeInteger

InnerModularGcd(R,BP,pMod,nextMod):C == T
 where
  R       :  EuclideanDomain
  BP      :  UnivariatePolynomialCategory(R)
  pMod    :  R
  nextMod :  (R,NNI) -> R

  C == with
     modularGcdPrimitive    : List BP  -> BP
         ++ modularGcdPrimitive(f1,f2) computes the gcd of the two polynomials
         ++ f1 and f2 by modular methods.
     modularGcd    : List BP  -> BP
         ++ modularGcd(listf) computes the gcd of the list of polynomials
         ++ listf  by modular methods.
     reduction :       (BP,R)              ->   BP
         ++ reduction(f,p) reduces the coefficients of the polynomial f
         ++ modulo the prime p.

  T == add

                    -- local functions --
    height    :         BP                ->   NNI
    mbound    :       (BP,BP)             ->   NNI
    modGcdPrimitive    :   (BP,BP)        ->   BP
    test         :     (BP,BP,BP)         ->   Boolean
    merge        :        (R,R)           ->   Union(R,"failed")
    modInverse   :      (R,R)             ->   R
    exactquo     :       (BP,BP,R)        ->   Union(BP,"failed")
    constNotZero :           BP           ->   Boolean
    constcase    : (List NNI ,List BP )   ->   BP 
    lincase      : (List NNI ,List BP )   ->   BP 


    if R has IntegerNumberSystem then
        reduction(u:BP,p:R):BP ==
            p = 0 => u
            map(symmetricRemainder(#1,p),u)
      else
        reduction(u:BP,p:R):BP ==
            p = 0 => u
            map(#1 rem p,u)

    FP:=EuclideanModularRing(R,BP,R,reduction,merge,exactquo)
    zeroChar : Boolean := R has CharacteristicZero

                 --  exported functions --

    -- modular Gcd for a list of primitive polynomials
    modularGcdPrimitive(listf : List BP) :BP ==
      empty? listf => 0$BP
      g := first listf
      for f in rest listf | not zero? f  while positive? degree g repeat
        g:=modGcdPrimitive(g,f)
      g		

    -- gcd for univariate polynomials
    modularGcd(listf : List BP): BP  ==
      listf:=remove!(0$BP,listf)
      empty? listf => 0$BP
      # listf = 1 => first listf 
      minpol:=1$BP
      -- extract a monomial gcd
      mdeg:= "min"/[minimumDegree f for f in listf]
      if positive? mdeg then
        minpol1:= monomial(1,mdeg)
        listf:= [(f exquo minpol1)::BP for f in listf]
        minpol:=minpol*minpol1
      listdeg:=[degree f for f in listf ]
    -- make the polynomials primitive
      listCont := [content f for f in listf]
      contgcd:= gcd listCont
      -- make the polynomials primitive
      listf :=[(f exquo cf)::BP for f in listf for cf in listCont]
      minpol:=contgcd*minpol
      ans:BP :=  
         --one polynomial is constant
         member?(1,listf) => 1
         --one polynomial is linear
         member?(1,listdeg) => lincase(listdeg,listf)
         modularGcdPrimitive listf
      minpol*ans
 
                  --  local functions --

    --one polynomial is linear, remark that they are primitive
    lincase(listdeg:List NNI ,listf:List BP ): BP  ==
      n:= position(1,listdeg)
      g:=listf.n
      for f in listf repeat
        if (f1:=f exquo g) case "failed" then return 1$BP
      g

    -- test if d is the gcd
    test(f:BP,g:BP,d:BP):Boolean ==
      d0:=coefficient(d,0)
      coefficient(f,0) exquo d0 case "failed" => false
      coefficient(g,0) exquo d0 case "failed" => false
      f exquo d case "failed" => false
      g exquo d case "failed" => false
      true

    -- gcd and cofactors for PRIMITIVE univariate polynomials
    -- also assumes that constant terms are non zero
    modGcdPrimitive(f:BP,g:BP): BP ==
      df:=degree f
      dg:=degree g
      dp:FP
      lcf:=leadingCoefficient f
      lcg:=leadingCoefficient g
      testdeg:NNI
      lcd:R:=gcd(lcf,lcg)
      prime:=pMod
      bound:=mbound(f,g)
      while zero? (lcd rem prime) repeat
        prime := nextMod(prime,bound)
      soFar:=gcd(reduce(f,prime),reduce(g,prime))::BP
      testdeg:=degree soFar
      zero? testdeg => return 1$BP
      ldp:FP:=
        ((lcdp:=leadingCoefficient(soFar::BP)) = 1) =>
                                        reduce(lcd::BP,prime)
        reduce((modInverse(lcdp,prime)*lcd)::BP,prime)
      soFar:=reduce(ldp::BP *soFar,prime)::BP
      soFarModulus:=prime
      -- choose the prime
      while true repeat
        prime := nextMod(prime,bound)
        lcd rem prime =0 => "next prime"
        fp:=reduce(f,prime)
        gp:=reduce(g,prime)
        dp:=gcd(fp,gp)
        dgp :=euclideanSize dp
        if dgp =0 then return 1$BP
        if dgp=dg and not (f exquo g case "failed") then return g
        if dgp=df and not (g exquo f case "failed") then return f
        dgp > testdeg => "next prime"
        ldp:FP:=
          ((lcdp:=leadingCoefficient(dp::BP)) = 1) =>
                                        reduce(lcd::BP,prime)
          reduce((modInverse(lcdp,prime)*lcd)::BP,prime)
        dp:=ldp *dp
        dgp=testdeg  =>
           correction:=reduce(dp::BP-soFar,prime)::BP
           zero? correction =>
              ans:=reduce(lcd::BP*soFar,soFarModulus)::BP
              cont:=content ans
              ans:=(ans exquo cont)::BP
              test(f,g,ans) => return ans
              soFarModulus:=soFarModulus*prime
           correctionFactor:=modInverse(soFarModulus rem prime,prime)
                             -- the initial rem is just for efficiency
           soFar:=soFar+soFarModulus*(correctionFactor*correction)
           soFarModulus:=soFarModulus*prime
           soFar:=reduce(soFar,soFarModulus)::BP
        dgp<testdeg =>
          soFarModulus:=prime
          soFar:=dp::BP
          testdeg:=dgp
        if not zeroChar and euclideanSize(prime)>1 then
           result:=dp::BP
           test(f,g,result) => return result
        -- this is based on the assumption that the caller of this package,
        -- in non-zero characteristic, will use primes of the form
        -- x-alpha as long as possible, but, if these are exhausted,
        -- will switch to a prime of degree larger than the answer
        -- so the result can be used directly.

    merge(p:R,q:R):Union(R,"failed") ==
         p = q => p
         p = 0 => q
         q = 0 => p
         "failed"

    modInverse(c:R,p:R):R ==
        (extendedEuclidean(c,p,1)::Record(coef1:R,coef2:R)).coef1

    exactquo(u:BP,v:BP,p:R):Union(BP,"failed") ==
        invlcv:=modInverse(leadingCoefficient v,p)
        r:=monicDivide(u,reduction(invlcv*v,p))
        not zero? reduction(r.remainder,p) => "failed"
        reduction(invlcv*r.quotient,p)


    -- compute the height of a polynomial --
    height(f:BP):NNI ==
      degf:=degree f
      "max"/[euclideanSize cc for cc in coefficients f]

    -- compute the bound
    mbound(f:BP,g:BP):NNI ==
      hf:=height f
      hg:=height g
      2*min(hf,hg)

-- ForModularGcd(R,BP) : C == T
--  where
--   R          :   EuclideanDomain  -- characteristic 0
--   BP         :   UnivariatePolynomialCategory(R)
--
--   C == with
--     nextMod :  (R,NNI) -> R
--
--   T == add
--     nextMod(val:R,bound:NNI) : R  ==
--       ival:Z:= val pretend Z
--       (nextPrime(ival)$IntegerPrimesPackage(Z))::R
--
-- ForTwoGcd(F) : C == T
--  where
--   F          :   Join(Finite,Field)
--   SUP       ==>  SparseUnivariatePolynomial
--   R         ==>  SUP F
--   P         ==>  SUP R
--   UPCF2     ==>  UnivariatePolynomialCategoryFunctions2
--
--   C == with
--     nextMod :  (R,NNI) -> R
--
--   T == add
--     nextMod(val:R,bound:NNI) : R ==
--       ris:R:= nextItem(val) :: R
--       euclideanSize ris < 2 => ris
--       generateIrredPoly(
--             (bound+1)::PositiveInteger)$IrredPolyOverFiniteField(F)
--
--
-- ModularGcd(R,BP) == T
--  where
--   R  : EuclideanDomain -- characteristic 0
--   BP : UnivariatePolynomialCategory(R)
--   T ==> InnerModularGcd(R,BP,67108859::R,nextMod$ForModularGcd(R,BP))
--
-- TwoGcd(F) : C == T
--  where
--   F          :   Join(Finite,Field)
--   SUP       ==>  SparseUnivariatePolynomial
--   R         ==>  SUP F
--   P         ==>  SUP R
--
--   T ==> InnerModularGcd(R,P,nextMod(monomial(1,1)$R)$ForTwoGcd(F),
--                         nextMod$ForTwoGcd(F))

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

<<package INMODGCD InnerModularGcd>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
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\end{document}