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SandBoxMultivariatePolynomialDivision
  
last edited 15 years ago by Bill Page

Edit detail for SandBoxMultivariatePolynomialDivision revision 1 of 3
1 2 3
Editor: Bill Page
Time: 2008/07/22 05:59:38 GMT-7
Note: new 
changed:
-
A tutorial written by Angelos Mantzaflaris (currently on in Greek)
 
- http://eclass.uoa.gr/courses/MATH117/document/Simeioseis_axiom.pdf

contains the following routine:
\begin{axiom}
-- Multivariate Polynomial Division
mpolydiv (f,D)==
  lex := typeOf (f)
  Q:=[ 0:: lex for j in D ]
  r:= 0:: lex
  m:= #D
  lms :=[ leadingMonomial (D.j) for j in 1..m]
  j:= 1
  flag := false
  while f ~= 0 repeat
    if j>m then
      j:= 1
      if flag = false then
        t:= leadingMonomial f
        r:= r + t
        f:= f - t
      flag := false
      if normalForm ( leadingMonomial (f),[ lms .j ])=0 then
        t := ( leadingMonomial f/lms.j):: lex
        Q.j := Q.j + t
        f := f - t*D.j
        flag := true
      else
        j:= j+1
  rec := Record ( quotient : LIST lex , remainder : lex)
  return [Q,r]:: rec
\end{axiom}

Here is an example of it's use:
\begin{axiom}
lex:=DMP([x,y,z], FRAC INT)
d1: lex := -2* x^12 + 4* x^2 * y^4
d2: lex := x^2 - y^11
f :lex := 3*x^14*y^10 + 5*x*y^3 -6*y^7 + 1
mpolydiv(f, [d1,d2])
mpolydiv(f, [d2,d1])
.quotient 1 * d2 + %.quotient 2 * d1 + %.remainder
\end{axiom}


A tutorial written by Angelos Mantzaflaris (currently on in Greek)



    

  • http://eclass.uoa.gr/courses/MATH117/document/Simeioseis_axiom.pdf
    • 




contains the following routine:
\begin{axiom}
-- Multivariate Polynomial Division
mpolydiv (f,D)==
  lex := typeOf (f)
  Q:=[ 0:: lex for j in D ]
  r:= 0:: lex
  m:= #D
  lms :=[ leadingMonomial (D.j) for j in 1..m]
  j:= 1
  flag := false
  while f ~= 0 repeat
    if j>m then
      j:= 1
      if flag = false then
        t:= leadingMonomial f
        r:= r + t
        f:= f - t
      flag := false
      if normalForm ( leadingMonomial (f),[ lms .j ])=0 then
        t := ( leadingMonomial f/lms.j):: lex
        Q.j := Q.j + t
        f := f - t*D.j
        flag := true
      else
        j:= j+1
  rec := Record ( quotient : LIST lex , remainder : lex)
  return [Q,r]:: rec
\end{axiom}


Here is an example of it's use:
\begin{axiom}
lex:=DMP([x,y,z], FRAC INT)
d1: lex := -2 x^12 + 4 x^2  y^4
d2: lex := x^2 - y^11
f :lex := 3x^14y^10 + 5xy^3 -6y^7 + 1
mpolydiv(f, [d1,d2])
mpolydiv(f, [d2,d1])
.quotient 1  d2 + %.quotient 2  d1 + %.remainder
\end{axiom}




    Some or all expressions may not have rendered properly,
    because Axiom returned the following error:
Error: export AXIOM=/usr/local/lib/axiom/target/x86_64-unknown-linux; ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=$ALDORROOT/bin:$PATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 240; $AXIOM/bin/AXIOMsys < /var/zope2/var/LatexWiki/7310038064502082312-25px.axm


GCL (GNU Common Lisp)  2.6.8 CLtL1    Nov  9 2007 07:47:56
Source License: LGPL(gcl,gmp), GPL(unexec,bfd,xgcl)
Binary License:  GPL due to GPL'ed components: (READLINE BFD UNEXEC)
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Use (help) to get some basic information on how to use GCL.
Temporary directory for compiler files set to /tmp/
                 FriCAS (AXIOM fork) Computer Algebra System 
                         Version: FriCAS 2007-10-02
               Timestamp: Friday November 9, 2007 at 19:35:06 
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------




(1) -> (1) -> (1) -> (1) -> (1) ->


Multivariate Polynomial Division
mpolydiv (f,D)==
  lex := typeOf (f)
  Q:=[ 0:: lex for j in D ]
  r:= 0:: lex
  m:= #D
  lms :=[ leadingMonomial (D.j) for j in 1..m]
  j:= 1
  flag := false
  while f ~= 0 repeat
    if j>m then
      j:= 1
      if flag = false then
        t:= leadingMonomial f
        r:= r + t
        f:= f - t
      flag := false
      if normalForm ( leadingMonomial (f),[ lms .j ])=0 then
        t := ( leadingMonomial f/lms.j):: lex
        Q.j := Q.j + t
        f := f - t*D.j
        flag := true
      else
        j:= j+1
  rec := Record ( quotient : LIST lex , remainder : lex)
  return [Q,r]:: rec
                                                                   Type: Void
(2) -> lex:=DMP([x,y,z], FRAC INT)






$$
\mbox{\rm DistributedMultivariatePolynomial([x,y,z],Fraction Integer)} 
\leqno(2)
$$
                                                                 Type: Domain
d1: lex := -2 x^12 + 4 x^2 * y^4




$$
-{2 \  {x \sp {12}}}+{4 \  {x \sp 2} \  {y \sp 4}} 
\leqno(3)
$$
            Type: DistributedMultivariatePolynomial([x,y,z],Fraction Integer)
d2: lex := x^2 - y^11




$$
{x \sp 2} -{y \sp {11}} 
\leqno(4)
$$
            Type: DistributedMultivariatePolynomial([x,y,z],Fraction Integer)
f :lex := 3x^14y^10 + 5xy^3 -6*y^7 + 1




$$
{3 \  {x \sp {14}} \  {y \sp {10}}}+{5 \  x \  {y \sp 3}} -{6 \  {y \sp 7}}+1 
\leqno(5)
$$
            Type: DistributedMultivariatePolynomial([x,y,z],Fraction Integer)
mpolydiv(f, [d1,d2])


   Cannot compile conversion for types involving local variables. In 
      particular, could not compile the expression involving :: lex 
   FriCAS will attempt to step through and interpret the code.
Some or all expressions may not have rendered properly, because Latex returned the following error:
This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
 \write18 enabled.
entering extended mode
(./4324228999094147848-18.0px.tex
LaTeX2e <2003/12/01>
Babel <v3.8d> and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf-tetex/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf-tetex/tex/latex/base/size12.clo))
(/usr/share/texmf-tetex/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf-tetex/tex/latex/amsmath/amstext.sty
(/usr/share/texmf-tetex/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf-tetex/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf-tetex/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf-tetex/tex/latex/amsfonts/amsfonts.sty)
(/usr/share/texmf-tetex/tex/latex/amsfonts/amssymb.sty)
(/usr/share/texmf-tetex/tex/latex/amscls/amsthm.sty)
(/usr/share/texmf-tetex/tex/generic/xypic/xy.sty
(/usr/share/texmf-tetex/tex/generic/xypic/xy.tex Bootstrap'ing: catcodes,
docmode, (/usr/share/texmf-tetex/tex/generic/xypic/xyrecat.tex)
(/usr/share/texmf-tetex/tex/generic/xypic/xyidioms.tex)


 Xy-pic version 3.7 <1999/02/16>
 Copyright (c) 1991-1998 by Kristoffer H. Rose <krisrose@ens-lyon.fr>
 Xy-pic is free software: see the User's Guide for details.




Loading kernel: messages; fonts; allocations: state, direction,
utility macros; pictures: \xy, positions, objects, decorations;
kernel objects: directionals, circles, text; options; algorithms: directions,
edges, connections;  Xy-pic loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xyall.tex
 Xy-pic option: All features v.3.3
(/usr/share/texmf-tetex/tex/generic/xypic/xycurve.tex
 Xy-pic option: Curve and Spline extension v.3.7 curve, circles, loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xyframe.tex
 Xy-pic option: Frame and Bracket extension v.3.7 loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xycmtip.tex
 Xy-pic option: Computer Modern tip extension v.3.3
(/usr/share/texmf-tetex/tex/generic/xypic/xytips.tex
 Xy-pic option: More Tips extension v.3.3 loaded) loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xyline.tex
 Xy-pic option: Line styles extension v.3.6 loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xyrotate.tex
 Xy-pic option: Rotate and Scale extension v.3.3 loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xycolor.tex
 Xy-pic option: Colour extension v.3.3 loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xymatrix.tex
 Xy-pic option: Matrix feature v.3.4 loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xyarrow.tex
 Xy-pic option: Arrow and Path feature v.3.5 path, \ar, loaded)
(/usr/share/texmf-tetex/tex/generic/xypic/xygraph.tex
 Xy-pic option: Graph feature v.3.7 loaded) loaded))
(/usr/share/texmf-tetex/tex/latex/graphics/graphicx.sty
(/usr/share/texmf-tetex/tex/latex/graphics/keyval.sty)
(/usr/share/texmf-tetex/tex/latex/graphics/graphics.sty
(/usr/share/texmf-tetex/tex/latex/graphics/trig.sty)
(/usr/share/texmf-tetex/tex/latex/graphics/graphics.cfg)
(/usr/share/texmf-tetex/tex/latex/graphics/dvips.def)))
(/usr/share/texmf-tetex/tex/latex/graphics/color.sty
(/usr/share/texmf-tetex/tex/latex/graphics/color.cfg)
(/usr/share/texmf-tetex/tex/latex/graphics/dvipsnam.def))
(/usr/share/texmf-tetex/tex/latex/tools/verbatim.sty)
(/usr/share/texmf-tetex/tex/latex/graphviz/graphviz.sty
(/usr/share/texmf-tetex/tex/latex/psfrag/psfrag.sty))
(/usr/share/texmf-tetex/tex/latex/sage/sagetex.sty
Writing sage input file 4324228999094147848-18.0px.sage
(./4324228999094147848-18.0px.sout))
(/usr/share/texmf-tetex/tex/latex/gnuplottex/gnuplottex.sty
(/usr/share/texmf-tetex/tex/latex/base/latexsym.sty)
(/usr/share/texmf-tetex/tex/latex/moreverb/moreverb.sty)
(/usr/share/texmf-tetex/tex/latex/base/ifthen.sty))
(./4324228999094147848-18.0px.aux) [1]
 Text line contains an invalid character.
l.97 ^^B
        .quotient 1  d2 + %.quotient 2  d1 + %.remainder
[2] (/usr/share/texmf-tetex/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf-tetex/tex/latex/amsfonts/umsb.fd)
(/usr/share/texmf-tetex/tex/latex/base/ulasy.fd) [3] [4] [5] [6] [7]
(./4324228999094147848-18.0px.aux) )
(see the transcript file for additional information)
Output written on 4324228999094147848-18.0px.dvi (7 pages, 2976 bytes).
Transcript written on 4324228999094147848-18.0px.log.