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Faugère F5 algorithm : ウィキペディア英語版
Faugère's F4 and F5 algorithms
In computer algebra, the Faugère F4 algorithm, by Jean-Charles Faugère, computes the Gröbner basis of an ideal of a multivariate polynomial ring. The algorithm uses the same mathematical principles as the Buchberger algorithm, but computes many normal forms in one go by forming a generally sparse matrix and using fast linear algebra to do the reductions in parallel.
The Faugère F5 algorithm first calculates the Gröbner basis of a pair of generator polynomials of the ideal. Then it uses this basis to reduce the size of the initial matrices of generators for the next larger basis:

If ''G''prev is an already computed Gröbner basis (''f''2, …, ''f''''m'') and we want to compute a Gröbner basis of (''f''1) + ''G''prev then we will construct matrices whose rows are ''m'' ''f''1 such that ''m'' is a monomial not divisible by the leading term of an element of ''G''prev.

This strategy allows the algorithm to apply two new criteria based on what Faugère calls ''signatures'' of polynomials. Thanks to these criteria, the algorithm can compute Gröbner bases for a large class of interesting polynomial systems, called ''regular sequences'', without ever simplifying a single polynomial to zero—the most time-consuming operation in algorithms that compute Gröbner bases. It is also very effective for a large number of non-regular sequences.
== Implementations ==

The Faugère F4 algorithm is implemented
* as a (package FGb ) for the Maple computer algebra system. This package is included in Maple distribution as the option method=fgb of function Groebner();
* in the Magma computer algebra system.
* as a (C library ).
Study versions of the Faugère F5 algorithm is implemented in
* the SINGULAR computer algebra system;
* the Sage computer algebra system.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Faugère's F4 and F5 algorithms」の詳細全文を読む



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