Changeset c403c6 in git


Ignore:
Timestamp:
Oct 1, 2010, 11:30:20 AM (14 years ago)
Author:
Viktor Levandovskyy <levandov@…>
Branches:
(u'spielwiese', '17f1d200f27c5bd38f5dfc6e8a0879242279d1d8')
Children:
4131d807c7d28a79ca0e9721b4af967315686e45
Parents:
9f5ebf25535e7f3b32e56ffc7635be3bcf1c05bf
Message:
*levandov: docu update for sagbi2

git-svn-id: file:///usr/local/Singular/svn/trunk@13359 2c84dea3-7e68-4137-9b89-c4e89433aadc
File:
1 edited

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  • Singular/LIB/sagbi2.lib

    r9f5ebf rc403c6  
    33category="Commutative Algebra";
    44info="
    5 LIBRARY: sagbi2.lib  Compute SAGBI basis (subalgebra bases analogous to Groebner bases for ideals) of a given subalgebra
     5LIBRARY: sagbi2.lib  Compute SAGBI basis (subalgebra bases analogous to Groebner bases for ideals) of a subalgebra
    66AUTHORS: Jan Hackfeld, Jan.Hackfeld@rwth-aachen.de
    77
    … …  
    1111 sagbi(A [,m,t]);      construct a SAGBI basis for the given subalgebra
    1212 sagbiPart(A,k[,m]);   constructs a partial SAGBI basis for the given subalgebra
    13 
     13 algebraicDependence(I,it);  compute it iteration of SAGBI for algebraic dependencies of I
     14
     15SEE ALSO: algebra_lib
    1416";
     17
     18
    1519//AUXILIARY PROCEDURES:
    1620//uniqueVariableName(s)                         adds character "@" at the beginning of string s until this is a unique new variable name
    1721//extendRing(r, ...)                                    creates a new ring, which is an extension of r
    18 //proc stdKernPhi(kernNew, kernOld,...) computes Groebner basis of kernNew+kernOld
     22//stdKernPhi(kernNew, kernOld,...) computes Groebner basis of kernNew+kernOld
    1923//spolynomialsGB(A,...)                                 computes the SAGBI S-polynomials of the subalgebra defined by the generators in A using Groebner bases
    2024//spolynomialsToric(A)                          computes the SAGBI S-polynomials of the subalgebra defined by the generators in A using toric.lib
    … …  
    423427
    424428proc sagbiSPoly(ideal algebra,list #)
    425   "USAGE:   sagbiSPoly(A[, returnRing, meth]);  A is an ideal, returnRing and meth are integers.
    426 RETURN:   Returns SAGBI S-polynomials of the leading terms of given ideal A if returnRing==0.
     429"USAGE:   sagbiSPoly(A[, returnRing, meth]);  A is an ideal, returnRing and meth are integers.
     430RETURN:   Returns SAGBI S-polynomials of the leading terms of given ideal A if returnRing=0.
    427431@*        Otherwise returns a new ring containing the ideals algebraicRelations
    428432@*        and spolynomials, where these objects are explained by their name.
    … …  
    491495  ring r= 0,(x,y),dp;
    492496  ideal A=x*y+x,x*y^2,y^2+y,x^2+x;
    493   //------------------ Computing the SAGBI S-polynomials only
     497  //------------------ Compute the SAGBI S-polynomials only
    494498  sagbiSPoly(A);
    495   //------------------ Now the extended ring is returned,
    496   // containing the ideal of algebraic relations and the ideal of the S-polynomials
     499  //------------------ Extended ring is to be returned, which contains
     500  // the ideal of algebraic relations and the ideal of the S-polynomials
    497501  def rNew=sagbiSPoly(A,1);  setring rNew;
    498502  spolynomials;
    … …  
    508512
    509513proc sagbiReduce(idealORpoly, ideal algebra, list #)
    510   "USAGE:   sagbiReduce(idealORpoly, A[, tailred, meth]]); idealORPoly is an ideal or poly, A is an ideal, tailred and meth are both integers
    511 RETURN:   Returns the remainder(s) of idealORPoly after SAGBI reduction by A of type ideal or poly (same as input type).
     514"USAGE:   sagbiReduce(I, A[, tr, mt]); I, A ideals, tr, mt optional integers
     515RETURN:   Returns the remainder(s) of I after SAGBI reduction by A
    512516@format
    513     The optional argument tailred determines if tail-reduction is done.
    514      - If (tailred==0), no tail-reduction is done.
    515      - If (tailred!=0), tail-reduction is done.
     517    The optional argument tr=tailred determines whether tail reduction will be performed.
     518     - If (tailred=0), no tail reduction is done.
     519     - If (tailred<>0), tail reduction is done.
    516520     The other optional argument meth determines which method is
    517521         used for Groebner basis computations.
    518          - If meth==0 (default), the procedure std is used.
    519          - If meth==1, the procedure slimgb is used.
     522         - If mt=0 (default), the procedure std is used.
     523         - If mt=1, the procedure slimgb is used.
    520524@end format             
    521525EXAMPLE:  example sagbiReduce; shows an example"
    … …  
    571575  poly p2=x^16+x^12*y^5+6*x^8*y^4+x^6+y^4+3;
    572576  ideal P=p1,p2;
    573  
     577  //---------------------------------------------
    574578  //SAGBI reduction of polynomial p1 by algebra A. Default call, that is, no tail-reduction is done.
    575579  sagbiReduce(p1,A);
    576  
     580  //---------------------------------------------
    577581  //SAGBI reduction of set of polynomials P by algebra A, now tail-reduction is done.
    578582  sagbiReduce(P,A,1);
    … …  
    580584
    581585proc sagbi(ideal algebra, list #)
    582   "USAGE:   sagbi(A[, tailred, meth]])   A is an ideal, tailred and meth are both integers
     586"USAGE:   sagbi(A[, tr, mt]); A ideal, tr, mt optional integers
    583587RETURN: A SAGBI basis for the subalgebra given by the generators in A.
    584588@format
    585     The optional argument tailred determines if tail reduction is done.
    586      - If (tailred==0), no tail-reduction is done.
    587      - If (tailred!=0), tail-reduction is done.
     589    The optional argument tr=tailred determines whether tail reduction will be performed.
     590     - If (tailred=0), no tail reduction is performed,
     591     - If (tailred<>0), tail reduction is performed.
    588592     The other optional argument meth determines which method is
    589593         used for Groebner basis computations.
    590          - If meth==0 (default), the procedure std is used.
    591          - If meth==1, the procedure slimgb is used.
     594         - If mt=0 (default), the procedure std is used.
     595         - If mt=1, the procedure slimgb is used.
    592596@end format
    593597EXAMPLE:  example sagbi; shows an example"
    … …  
    624628  //Default call, no tail-reduction is done.
    625629  sagbi(A);
    626  
     630  //---------------------------------------------
    627631  //Call with tail-reduction and method specified.
    628632  sagbi(A,1,0);
    629633}
    630634
    631 
    632635proc sagbiPart(ideal algebra, int iterations, list #)
    633   "USAGE:   sagbiPart(A, k,[tailred, meth]]); A is an ideal, k, integer, tailred and meth are both integers
     636"USAGE:   sagbiPart(A, k,[tr, mt]); A is an ideal, k, tr and mt are integers
    634637RETURN: Performs k iterations of the SAGBI construction algorithm for the subalgebra given by the generators in A.
    635638@format
    636      The optional argument tailred determines if tailreduction is done.
    637      - If (tailred==0), no tail-reduction is done.
    638      - If (tailred!=0), tail-reduction is done.
     639     The optional argument tr=tailred determines if tail reduction will be performed.
     640     - If (tailred=0), no tail reduction is performed,
     641     - If (tailred<>0), tail reduction is performed.
    639642     The other optional argument meth determines which method is
    640643         used for Groebner basis computations.
    641          - If meth==0 (default), the procedure std is used.
    642          - If meth==1, the procedure slimgb is used.
     644         - If mt=0 (default), the procedure std is used.
     645         - If mt=1, the procedure slimgb is used.
    643646@end format
    644647EXAMPLE:  example sagbiPart; shows an example"
    … …  
    677680  //The following algebra does not have a finite SAGBI basis.
    678681  ideal A=x,xy-y2,xy2;
    679  
    680   //Call with two iterations, not tail-reduction is done.
     682  //---------------------------------------------------
     683  //Call with two iterations, no tail-reduction is done.
    681684  sagbiPart(A,2);
    682  
     685  //---------------------------------------------------
    683686  //Call with three iterations, tail-reduction and method 0.
    684687  sagbiPart(A,3,1,0);
    … …  
    686689
    687690
    688 //TODO: 1nsen als werden noch mit ausgegeben, aus algebraischen AbhÀngigkeiten entfernen
    689 // TODO2: finish the documentation
     691//DONE? 1nsen als werden noch mit ausgegeben, aus algebraischen AbhÀngigkeiten entfernen
     692// VL: finished the documentation
     693// TO comapre with algDependent from algebra_lib
    690694proc algebraicDependence(ideal I,int iterations)
     695"USAGE: algebraicDependence(I,it); I an an ideal, it is an integer
     696RETURN: In 'it' iterations, compute algebraic dependencies between elements of I
     697EXAMPLE: example algebraicDependence; shows an example"
    691698{
    692699  def br=basering;
    … …  
    782789  return(extendVarRing);
    783790}
     791example
     792{ "EXAMPLE:"; echo = 2;
     793  ring r= 0,(x,y),dp;
     794  //The following algebra does not have a finite SAGBI basis.
     795  ideal I=x^2, xy-y2, xy2;
     796  //---------------------------------------------------
     797  //Call with two iterations
     798  def DI = algebraicDependence(I,2);
     799  setring DI; algDep;
     800  // we see that no dependency has been seen so far
     801  //---------------------------------------------------
     802  //Call with two iterations
     803  setring r; kill DI;
     804  def DI = algebraicDependence(I,3);
     805  setring DI; algDep;
     806  map F = DI,x,y,x^2, xy-y2, xy2;
     807  F(algDep); // we see that it is a dependence indeed
     808}
     809
     810/*
     811  ring r= 0,(x,y),dp;
     812  //The following algebra does not have a finite SAGBI basis.
     813  ideal J=x^2, xy-y2, xy2, x^2*(x*y-y^2)^2 - (x*y^2)^2*x^4 + 11;
     814  //---------------------------------------------------
     815  //Call with two iterations
     816  def DI = algebraicDependence(J,2);
     817  setring DI; algDep;
     818*/
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