
TL;DR
This paper introduces an LDU matrix factorization algorithm over commutative domains that ensures numerical stability and is suitable for parallel computing environments, extending traditional LU methods.
Contribution
It generalizes the LEU-factorization to commutative domains, providing an error-free, stable, and parallelizable matrix decomposition method.
Findings
The algorithm decomposes matrices into L, D, U over commutative domains.
It avoids numerical instability present in previous methods.
The method is suitable for distributed memory supercomputers.
Abstract
LU-factorization of matrices is one of the fundamental algorithms of linear algebra. The widespread use of supercomputers with distributed memory requires a review of traditional algorithms, which were based on the common memory of a computer. Matrix block recursive algorithms are a class of algorithms that provide coarse-grained parallelization. The block recursive LU factorization algorithm was obtained in 2010. This algorithm is called LEU-factorization. It, like the traditional LU-algorithm, is designed for matrices over number fields. However, it does not solve the problem of numerical instability. We propose a generalization of the LEU algorithm to the case of a commutative domain and its field of quotients. This LDU factorization algorithm decomposes the matrix over the commutative domain into a product of three matrices, in which the matrices L and U belong to the commutative…
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Taxonomy
TopicsMatrix Theory and Algorithms · Distributed and Parallel Computing Systems · Coding theory and cryptography
