FastLinAlg -- faster linear algebra in certain cases

Description

FastLinAlg is a for computing (applications of) function field linear algebra more quickly in certain settings.

This package is still experimental. It provides functionality for doing certain linear algebra operations in function fields quickly. There is some multithreaded capability which is disabled by default.

Useful functions:

chooseGoodMinors Tries to find interesting minors of a matrix.
isRankAtLeast Tries to show that a matrix has rank at least a given number by looking at submatrices
regularInCodimension checks whether a ring is regular in codimension n only in the affirmative.
projDim checks the projective dimension of a module and may give better answers than pdim in the case that R is not homogeneous
recursiveMinors provides a different strategy for computing minors of a matrix. It is a cofactor strategy where the determinants of smaller minors are stored.
isCodimAtLeast provides a way for finding lower bounds for the codimension of an ideal, without actually computing the codimension.

Many of these functions have extensive options for fine tuning their behavior, for instance by controlling how submatrices are chosen. See the documentation for StrategyDefault

Acknowledgements:

The authors would like to thank David Eisenbud, Eloisa Grifo, and Srikanth Iyengar for useful conversations and comments on the development of this package.

Boyana Martinova received funding from the University of Utah Mathematics Department REU program and from the ACCESS program at the University of Utah, while developing this package.
Marcus Robinson received funding from the NSF RTG grant 1246989 while developing this package.
Karl Schwede received funding from NSF grant 1801849 while developing this package.
Yuhui (Wei) Yao received funding from the University of Utah Mathematics Department REU program, while developing this package

Authors

Boyana Martinova <u1056124@utah.edu>
Marcus Robinson <robinson@math.utah.edu>
Karl Schwede <schwede@math.utah.edu>
Yuhui (Wei) Yao <yuhuiyao4ever@gmail.com>

Version

This documentation describes version 0.21 of FastLinAlg.

Source code

The source code from which this documentation is derived is in the file FastLinAlg.m2.

Exports

Functions and commands
- chooseGoodMinors -- returns an ideal generated by interesting minors in a matrix
- chooseRandomSubmatrix -- returns coordinates for a random submatrix
- chooseSubmatrixLargestDegree -- returns coordinates for higher degree submatrix of a matrix
- chooseSubmatrixSmallestDegree -- returns coordinates for low degree submatrix of a matrix
- getSubmatrixOfRank -- tries to find a submatrix of the given rank
- isCodimAtLeast -- returns true if we can quickly see whether the codim is at least a given number
- isDimAtMost -- returns true if we can quickly see if the dim is at most a given number
- isRankAtLeast -- determines if the matrix has rank at least a number
- projDim -- finds an upper bound for the projective dimension of a module
- recursiveMinors -- uses a recursive cofactor algorithm to compute the ideal of minors of a matrix
- regularInCodimension -- attempts to show that the ring is regular in codimension n
- reorderPolynomialRing -- produces an isomorphic polynomial ring with a different, randomized, monomial order
Symbols
- PeriodicCheckFunction, see chooseGoodMinors -- returns an ideal generated by interesting minors in a matrix
- DetStrategy -- DetStrateg is a strategy for allowing the user to choose how determinants (or rank), is computed
- Rank, see DetStrategy -- DetStrateg is a strategy for allowing the user to choose how determinants (or rank), is computed
- Recursive, see DetStrategy -- DetStrateg is a strategy for allowing the user to choose how determinants (or rank), is computed
- GRevLexLargest -- strategies for choosing submatrices
- GRevLexSmallest, see GRevLexLargest -- strategies for choosing submatrices
- GRevLexSmallestTerm, see GRevLexLargest -- strategies for choosing submatrices
- LexLargest, see GRevLexLargest -- strategies for choosing submatrices
- LexSmallest, see GRevLexLargest -- strategies for choosing submatrices
- LexSmallestTerm, see GRevLexLargest -- strategies for choosing submatrices
- Random, see GRevLexLargest -- strategies for choosing submatrices
- RandomNonzero, see GRevLexLargest -- strategies for choosing submatrices
- SPairsFunction, see isCodimAtLeast -- returns true if we can quickly see whether the codim is at least a given number
- MaxMinors -- an option to control depth of search
- MinDimension -- an option for projDim
- ModP -- an option for regularInCodimension
- MinorsCache, see recursiveMinors -- uses a recursive cofactor algorithm to compute the ideal of minors of a matrix
- CodimCheckFunction, see regularInCodimension -- attempts to show that the ring is regular in codimension n
- MinMinorsFunction, see regularInCodimension -- attempts to show that the ring is regular in codimension n
- UseOnlyFastCodim, see regularInCodimension -- attempts to show that the ring is regular in codimension n
- Threads -- an option for various functions
Other things
- StrategyCurrent -- strategies for choosing submatrices
- StrategyDefault -- strategies for choosing submatrices
- StrategyDefaultNonRandom -- strategies for choosing submatrices
- StrategyGRevLexSmallest -- strategies for choosing submatrices
- StrategyLexSmallest -- strategies for choosing submatrices
- StrategyRandom -- strategies for choosing submatrices