Summary of algorithms from the Gröbner bases book and corresponding JAS classes and methods.
The JAS base package edu.jas name is omitted in the
following table.
A short explanation of code organization with interfaces and several implementing classes
can be found in the API guide.
| GB book algorithm | JAS interfaces, classes and methods | remarks |
0.1 DIVINT |
arith.BigInteger.divideAndRemainder
|
facade for java.math.BigInteger.divideAndRemainder
|
2.1 DIV |
structure.MonoidElem.divide
and
structure.MonoidElem.remainder
|
all classes which implement this interface |
2.2 DIVPOL |
poly.GenPolynomial.divideAndRemainder
|
for univariate polynomials over fields |
2.3 EXTEUC |
poly.GenPolynomial.egcd
|
for univariate polynomials over fields |
3.1 LINDEP |
|
|
3.2 EXCHANGE |
|
|
4.1 EQUIV |
structure.Residue.equals
or
application.Residue.equals
|
arbitrary residue class rings and residue class rings modulo polynomial ideals |
5.1 REDPOL |
gb.Reduction.normalform,
gb.ReductionAbstract.normalform,
gb.ReductionSeq.normalform
|
interface and sequential computation class, the method exists also for
polynomial lists and with a reduction recording matrix (this is exactly
REDPOL)
|
5.2 REDUCTION |
gb.Reduction.irreducibleSet
|
|
5.3 GRÖBNERTEST |
gb.GroebnerBase.isGB
|
provided by all classes which implement this interface |
5.4 GRÖBNER |
not implemented | |
5.5 REDGRÖBNER |
not implemented | |
5.6 GRÖBNERNEW1 |
gb.GroebnerBase.GB
|
provided by all classes which implement the interface |
5.7 UPDATE, 5.8 GRÖBNERNEW2 |
not implemented | |
5.9 EXTGRÖBNER |
gb.GroebnerBase.extGB
|
provided by some classes which implement the interface |
6.1 ELIMINATION |
application.Ideal.eliminate
|
the version with the String[] parameter only computes a Gröbner base wrt. the
respective elimination order
|
6.2 PROPER |
application.Ideal.isONE
|
proper ideal test is ! id.isONE()
|
6.3 INTERSECTION |
application.Ideal.intersect
|
for lists of ideals a simple iterative algorithm is used and for a pair of ideals it is the same algorithm |
6.4 CRT |
|
|
6.5 IDEALDIV1 |
application.Ideal.quotient
|
for an ideal a simple iterative algorithm is used and for one polynomial it is an algorithm without computing syzygies |
6.6 IDEALDIV2 |
application.Ideal.infiniteQuotientRab
and
application.Ideal.infiniteQuotientExponent
|
the exponent is computed in a separate step at the moment |
6.7 RADICALMEMTEST |
application.Ideal.isRadicalMember
|
the exponent is not computed |
6.8 SUBRINGMEMTEST |
|
|
8.1 PREDEC |
application.Ideal.zeroDimDecomposition
|
univariate polynomials of minimal degree in the ideal are irreducible
and not a power of an irreducible polynomial as specified in PREDEC
|
8.2 ZRADICALTEST |
application.Ideal.isZeroDimRadical
|
will also work in characteritsic p > 0 |
8.3 ZRADICAL |
application.Ideal.radical
|
ZRADICAL is containted as special case,
see also application.Ideal.zeroDimRadicalDecomposition
|
8.4 NORMPRIMDEC |
application.Ideal.zeroDimPrimaryDecomposition
|
contains all preprocessing steps, see ZPRIMDEC
|
8.5 NORMPOS |
application.Ideal.normalPositionFor
|
one step of NORMPOS as explained for the modified algorithm on page 383ff
|
8.6 ZPRIMDEC |
application.Ideal.zeroDimPrimaryDecomposition
|
returns a list of PrimaryComponent containers
|
8.7 CONT |
application.Ideal.contraction
|
more complicated since the permutation of variables must be considered also,
see application.Ideal.permContraction,
the polynomial f is returned in the IdealWithUniv container
|
8.8 EXTCONT |
application.Ideal.extension
|
only EXT, the combination EXTCONT is not implemented explicitly,
the polynomial f is returned in the IdealWithUniv container
|
8.9 RADICAL |
application.Ideal.radical
|
for ideals with arbitrary dimension |
8.10 PRIMDEC |
application.Ideal.primaryDecomposition
|
for ideals with arbitrary dimension |
8.11 VARSIGN |
root.RootUtil.signVar
|
|
8.12 STURMSEQ |
root.RealRootsSturm.sturmSequence
|
|
8.13 ISOLATE |
root.RealRoots.realRoots
|
could also be used for real root isolations not using Sturm sequences |
8.14 ISOREC |
root.RealRootsSturm.realRoots
|
interval bi-section until a root is isolated |
8.15 ISOREFINE |
root.RealRoots.refineInterval
|
also for real root isolations not using Sturm sequences |
8.16 SQUEEZE |
root.RealRoots.invariantMagnitudeInterval
|
see also
root.RealRoots.realMagnitude
|
8.17 REALZEROS |
application.PolyUtilApp.realAlgebraicRoots
|
different algorithm and different result data structure with RealAlgebraicNumbers
|
9.1 REDTERMS |
|
|
9.2 UNIVPOL |
application.Ideal.constructUnivariate
|
|
9.3 CONVGRÖBNER |
|
|
9.4 LMINTERM |
|
|
9.5 STRCONST |
|
|
9.6 DIMENSION |
application.Ideal.dimension
|
|
10.1 D-GRÖBNER |
gb.DGroebnerBaseSeq.GB
and
gb.EGroebnerBaseSeq.GB
|
|
Last modified: Thu Jun 17 23:09:17 CEST 2010