Algorithms used by Groebner[Basis] - Maple Help

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Algorithms used by Groebner[Basis]

Calling Sequence

Basis(J, tord, method=meth)

Parameters

J

-

a list or set of polynomials or a PolynomialIdeal

tord

-

a MonomialOrder, a ShortMonomialOrder, or a name

meth

-

one of the methods described below

Description

• 

The Groebner[Basis] command currently relies on a combination of five different algorithms to compute Groebner bases for various monomial orders and domains.  This help page documents these algorithms and their relative performance characteristics to help you decide what algorithm to use should the default choice prove unsatisfactory.  The algorithm is specified using an optional argument method=meth.

• 

method=fgb runs FGb, which is a compiled implementation of the F4 algorithm written by J.C. Faugere. FGb supports rational number and integer mod p coefficients, where p < 2^(kernelopts(wordsize)/2), which is 65536 on a 32-bit machine.  FGb supports tdeg orders and lexdeg orders with two blocks of variables.  It does not support parameters (that is, coefficients in a rational function field) or the output=extended option.

• 

method=maplef4 runs a Maple implementation of the F4 algorithm.  This implementation supports all monomial orders and coefficient fields as well as computations in non-commutative Ore algebras.  It is an exact implementation, meaning that it does not use modular methods, so it is preferable to run FGb if possible.  The output=extended option is currently not supported.

• 

method=buchberger runs the traditional Maple implementation of the Buchberger algorithm. This algorithm supports all domains, monomial orders and options such as output=extended. It is generally outperformed by F4 in most situations, but one notable exception is when a polynomial has been added to a Groebner basis and the basis is being recomputed. The implementation is based on Gebauer and Moller and uses the normal selection strategy.

• 

method=fglm runs the Groebner basis conversion algorithm of Faugere, Gianni, Lazard, and Mora. This algorithm supports all commutative domains, but requires the ideal to be zero-dimensional. See the Groebner[FGLM] help page for more details.  When this method is specified, Groebner[Basis] will choose one of the known Groebner bases (or compute one if none are known) and run Groebner[FGLM].

• 

method=walk runs the Groebner Walk conversion algorithm of Collart, Kalkbrener, and Mall, which supports all commutative domains and monomial orders.   See the Groebner[Walk] help page for more details.  When this method is specified, Groebner[Basis] will choose one of the known Groebner bases (or compute one if none are known) and run Groebner[Walk].  The Groebner walk is typically slower than FGLM for zero-dimensional ideals.

• 

In addition to the algorithms above, you can specify an overall strategy using the method=... option and Groebner[Basis] will choose whatever algorithms are appropriate.  The strategies are as follows:

• 

method=direct runs the fastest general purpose direct method.  This is currently FGb or Maple's F4 if FGb cannot be used. As better algorithms are implemented, this strategy will be updated to reflect the state-of-the-art method. Thus, calling Groebner[Basis] with method=direct is a way to ensure that your code always uses the fastest direct algorithm; even on future Maple releases.

• 

method=convert runs the fastest applicable conversion method, which is either FGLM or the Groebner Walk, depending on the monomial order and whether the ideal is zero-dimensional. This strategy is recommended for monomial orders that eliminate variables.  For 'plex', 'lexdeg' and 'prod' orders, the strategy is to run FGLM if possible.  For all other orders and for non-zero-dimensional ideals, the Groebner Walk is used.

• 

method=default runs the default strategy, which is subject to change in the future. Currently this is method=direct for 'tdeg', 'wdeg', 'grlex' and 'matrix' orders and non-commutative problems, and method=convert for 'plex', 'lexdeg' and 'prod' orders in the commutative case.

• 

Setting infolevel[GroebnerBasis] to a value between 1 and 5 directs all of the algorithms to print information about their progress and performance.

Examples

withGroebner&colon;

infolevel&lsqb;GroebnerBasis&rsqb;:=2&colon;

F:=x22xz&plus;5&comma;xy2&plus;yz3&comma;3y28z3

F:=x22xz&plus;5&comma;yz3&plus;xy2&comma;8z3&plus;3y2

(1)

BasisF&comma;tdegx&comma;y&comma;z

-> FGb
 total time:         0.023 sec

x22xz&plus;5&comma;8z33y2&comma;8xy2&plus;3y3&comma;9y4&plus;48y3z&plus;320y2

(2)

BasisF&comma;grlexx&comma;y&comma;z

-> F4 algorithm
 total time:         0.004 sec

x22xz&plus;5&comma;8z33y2&comma;8xy2&plus;3y3&comma;9y4&plus;48y3z&plus;320y2

(3)

BasisF&comma;plexx&comma;y&comma;z

-> FGLM algorithm
 total monomials  :   23
 total time   (sec)   .4e-2

9z996z8&plus;240z6&plus;1600z3&comma;3z8&plus;32z740z5&plus;80yz3&comma;8z3&plus;3y2&comma;9z896z7&plus;120z5&plus;640xz3&comma;x22xz&plus;5

(4)

IsZeroDimensionalF

true

(5)

BasisF&comma;lexdegx&comma;y&comma;z&comma;method&equals;walk

-> Groebner Walk
 total time:         0.000 sec

8z33y2&comma;9y4&plus;48y3z&plus;320y2&comma;8xy2&plus;3y3&comma;x22xz&plus;5

(6)

BasisF&comma;lexdegx&comma;y&comma;z&comma;method&equals;direct

-> FGb
 total time:         0.004 sec

9z996z8&plus;240z6&plus;1600z3&comma;3z8&plus;32z740z5&plus;80yz3&comma;9z896z7&plus;120z5&plus;640xz3&comma;8z3&plus;3y2&comma;x22xz&plus;5

(7)

BasisF&comma;plexz&comma;y&comma;x&comma;method&equals;direct

-> F4 algorithm
 total time:         0.017 sec

3x1264x11&plus;90x10960x9&plus;1125x84800x7&plus;7500x68000x5&plus;28125x4&plus;56250x2&plus;46875&comma;8x7&plus;3x6y&plus;120x5&plus;45x4y&plus;600x3&plus;225x2y&plus;1000x&plus;375y&comma;9x11&plus;192x10255x9&plus;2560x82925x7&plus;9600x616875x548750x340000x2&plus;5625y256250x&comma;3x1164x10&plus;90x9960x8&plus;1125x74800x6&plus;7500x58000x4&plus;28125x3&plus;46875x&plus;18750z

(8)

BasisF&comma;wdeg1&comma;2&comma;3&comma;x&comma;y&comma;z&comma;method&equals;convert

-> Groebner Walk
 total time:         0.033 sec

x2&plus;2xz5&comma;x43xy2&plus;10x2&plus;20z2&plus;25&comma;8xy2&plus;3y3&comma;x5&plus;3x2y215x350x50z&comma;8x6&plus;3x5y&plus;120x4&plus;45x3y&plus;600x2&plus;150xy&plus;150yz&plus;1000&comma;3x864x7&plus;90x6960x5&plus;900x44800x3&plus;450xy2&plus;450y2z&plus;3750x28000x&plus;5625

(9)

Non-commutative examples (see Basis_details for more information)

withOre_algebra&colon;

infolevel&lsqb;GroebnerBasis&rsqb;:=4&colon;

N:=3&colon;

A:=diff_algebraseqDi&comma;xi&comma;i&equals;1..N&comma;comm&equals;a&comma;b&comma;seqci&comma;i&equals;1..N&comma;polynom&equals;seqxi&comma;i&equals;1..N&colon;

S:=`+`seqxiDi&comma;i&equals;1..N&colon;

P:=skew_productS&plus;a&comma;S&plus;b&comma;A&colon;

F:=seqskew_productDi&comma;xiDi&plus;ci1&comma;AP&comma;i&equals;1..N

F:=D12x1&plus;D1c1D12x122D1D2x1x22D1D3x1x3a&plus;b&plus;1x1D1D22x222D2D3x2x3a&plus;b&plus;1x2D2D32x32a&plus;b&plus;1x3D3ab&comma;D22x2&plus;D2c2D12x122D1D2x1x22D1D3x1x3a&plus;b&plus;1x1D1D22x222D2D3x2x3a&plus;b&plus;1x2D2D32x32a&plus;b&plus;1x3D3ab&comma;D32x3&plus;D3c3D12x122D1D2x1x22D1D3x1x3a&plus;b&plus;1x1D1D22x222D2D3x2x3a&plus;b&plus;1x2D2D32x32a&plus;b&plus;1x3D3ab

(10)

tord:=tdegseqDi&comma;i&equals;1..N&comma;seqxi&comma;i&equals;1..N

tord:=tdegD1&comma;D2&comma;D3&comma;x1&comma;x2&comma;x3

(11)

T:=MonomialOrderA&comma;tord&colon;

G:=BasisF&comma;T&colon;

-> F4 algorithm
 domain: SkewAlgebra   coeffs: rat_fcn
 basis: 1 of 3, syzygies: 3 of 3, degree: 4
 0 x 0 with 3 rhs
 basis: 2 of 5, syzygies: 1 of 3, degree: 4
 2 x 2 with 1 rhs
 basis: 3 of 6, syzygies: 2 of 2, degree: 5
 4 x 4 with 2 rhs
 basis: 5 of 8, syzygies: 4 of 4, degree: 6
 12 x 12 with 4 rhs
 basis: 8 of 11, syzygies: 11 of 12, degree: 7
 39 x 39 with 11 rhs
 basis: 10 of 13, syzygies: 10 of 10, degree: 8
 75 x 75 with 10 rhs
 basis: 12 of 15, syzygies: 7 of 7, degree: 9
 94 x 94 with 7 rhs
 symbolic prc        0.154 sec
 inter reduce        0.003 sec
 update pairs        0.000 sec
 reduce basis        0.010 sec
 total time:         0.167 sec
-----------------------------

Groebner:-LeadingMonomialG&comma;T

D22x2&comma;D12x1&comma;D1D2x1x2&comma;D1D32x1x3&comma;D1D2D3x1x3&comma;x2D2D32x1x3&comma;D12D2D3x2x3&comma;D2D33x1x3&comma;D1D2D32x22x3&comma;D12D33x2x3&comma;D34x12x32&comma;D12D22D32x32

(12)

See Also

Basis, Basis_details, LeadingMonomial, MonomialOrder

References

  

Buchberger, B. "Grobner Bases: An algorithmic method in polynomial ideal theory." In Multidimensional systems theory, pp. 184-232, Reidel, 1985.

  

Collart, S.; Kalkbrener, M.; and Mall, D. "Converting Bases with the Grobner Walk."  J. Symbolic Comput., Vol. 3, No. 4, (1997): 465-469.

  

Faugere, J. "A New Efficient Algorithm for Computing Grobner Bases (F4)." Journal of Pure and Applied Algebra, Vol. 139, No. 1-3, (1999): 61-88.

  

Faugere, J.; Gianni, P.; Lazard, D.; and Mora, T. "Efficient computation of zero-dimensional Grobner bases by change of ordering." J. Symbolic Comput., Vol. 16, (1993): 329-344.

  

Gebauer, R., and Moller, H. "On an installation of Buchberger's algorithm." J. Symbolic Comput., Vol. 6, (1988): 275-286.

  

Pearce, R., and Monagan, M. "A Sparse Algorithm for Polynomial Division with Application to F4." in preparation, 2006.


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