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LSU factorization

Gennadi Malaschonok

TL;DR

The paper generalizes LU factorization to LSU for matrices over a commutative domain and its quotient field by constructing a sparse factor $S$ from surrounding minors, achieving $A=LSU$ with $L$ lower and $U$ upper triangular. It develops a dichotomous, block-recursive algorithm that also computes auxiliary factors $M$ and $W$ and the transformed $\widehat{S}$, yielding a pseudoinverse when $A$ is singular. The authors prove correctness, derive a matrix-multiplication–level complexity $t(n)\sim \sigma n^{\beta}$ (with $\sigma$ depending on constants and the field size) and illustrate the method with a detailed $4\times4$ example. They argue LSU is particularly advantageous for sparse matrices and for matrices over integers or polynomials where traditional LU can be less effective or inapplicable.

Abstract

The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm decomposes any matrix A into a product of three matrices A=LSU, where each element of the triangular matrices L and U is a minor of matrix A. The number of non-zero elements in matrix S is equal to rank(A), and each of them is the inverse of the product of a specific pair of matrix A minors. The algorithm's complexity is equivalent to that of matrix multiplication.

LSU factorization

TL;DR

The paper generalizes LU factorization to LSU for matrices over a commutative domain and its quotient field by constructing a sparse factor from surrounding minors, achieving with lower and upper triangular. It develops a dichotomous, block-recursive algorithm that also computes auxiliary factors and and the transformed , yielding a pseudoinverse when is singular. The authors prove correctness, derive a matrix-multiplication–level complexity (with depending on constants and the field size) and illustrate the method with a detailed example. They argue LSU is particularly advantageous for sparse matrices and for matrices over integers or polynomials where traditional LU can be less effective or inapplicable.

Abstract

The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm decomposes any matrix A into a product of three matrices A=LSU, where each element of the triangular matrices L and U is a minor of matrix A. The number of non-zero elements in matrix S is equal to rank(A), and each of them is the inverse of the product of a specific pair of matrix A minors. The algorithm's complexity is equivalent to that of matrix multiplication.

Paper Structure

This paper contains 16 sections, 3 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Initial statement of the problem
  3. Preliminaries
  4. Semigroup of weighted permutations
  5. Some mappings on semigroups
  6. Surrounding minors
  7. Statement of the problem
  8. General statement of the problem
  9. Inverse and pseudo inverse matrices
  10. Construction of a dichotomous recursive factorization
  11. Algorithm of LSU factorization
  12. Complexity
  13. The number of arithmetic operations in the numeric field
  14. Examples
  15. Detailed description of an example of matrix factorization
  16. ...and 1 more sections

Key Result

Theorem 1

Let $A$ be a square matrix, $A^*$ the adjoint matrix for $A$, $det (A) \ne 0$, $G$ (1) be the surrounding matrix for $A$, then

Figures (1)

  • Figure 1: The structure of the weighted permutations semigroup ${\bf S}_{wp}$

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Theorem 2
  • Theorem 3