LDU factorization

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 domain, and the elements of the weighted truncated permutation matrix D are the elements inverse to the product of some pair of minors. All elements are calculated without errors, so the problem of instability does not arise.