Numerical Linear Algebra for High Performance Computers
Numerical Linear Algebra for High Performance Computers
On parallel block algorithms for exact triangularizations
Parallel Computing
Dense Linear Algebra over Word-Size Prime Fields: the FFLAS and FFPACK Packages
ACM Transactions on Mathematical Software (TOMS)
LU factoring of non-invertible matrices
ACM Communications in Computer Algebra
Fast generalized bruhat decomposition
CASC'10 Proceedings of the 12th international conference on Computer algebra in scientific computing
Multiplying matrices faster than coppersmith-winograd
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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Gaussian elimination with full pivoting generates a PLUQ matrix decomposition. Depending on the strategy used in the search for pivots, the permutation matrices can reveal some information about the row or the column rank profiles of the matrix. We propose a new pivoting strategy that makes it possible to recover at the same time both row and column rank profiles of the input matrix and of any of its leading sub-matrices. We propose a rank-sensitive and quad-recursive algorithm that computes the latter PLUQ triangular decomposition of an m x n matrix of rank r in O(mnrω-2) field operations, with ω the exponent of matrix multiplication. Compared to the LEU decomposition by Malashonock, sharing a similar recursive structure, its time complexity is rank sensitive and has a lower leading constant. Over a word size finite field, this algorithm also improves the practical efficiency of previously known implementations.