LSU Factorization
Year: 2023
Type: conference-paper
Abstract: Matrix LU factorization algorithm is one of the fundamental algorithms of linear algebra. However, the need to perform calculations with large matrices and use distributed memory supercomputers requires a revision of traditional algorithms. 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 LSU factorization algorithm decomposes any matrix A over a commutative domain into a product of three matrices, where each element of the triangular matrices L and U is a minor of matrix A.The number of non-zero elements of the matrix $S$ is equal to rank(A) and each of them is the inverse of the product of some pair of matrix A minors. All these matrices can be calculated without errors, in which case there will be no problem of computational instability. In the case of a finite field, the algorithm has the complexity of matrix multiplication, and in the case of integer matrices of size n, the complexity in the number of bit operations increases by n times.
Author: Gennadi Malaschonok
Institution: National University of Kyiv-Mohyla Academy
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4400811132