Matrix factoring by fraction-free reduction

TL;DR

This paper studies exact matrix decomposition using Gauss-Bareiss reduction, focusing on common factors, pivoting strategies, and their effects on the output, supported by experimental data.

Contribution

It identifies systematic and statistical common factors in fraction-free matrix decompositions and analyzes how pivoting strategies influence the results.

Findings

Existing algorithms introduce specific common factors.

Pivoting strategies affect the size of output factors.

Experimental data supports theoretical insights.

Abstract

We consider exact matrix decomposition by Gauss-Bareiss reduction. We investigate two aspects of the process: common row and column factors and the influence of pivoting strategies. We identify two types of common factors: systematic and statistical. Systematic factors depend on the process, while statistical factors depend on the specific data. We show that existing fraction-free QR (Gram-Schmidt) algorithms create a common factor in the last column of Q. We relate the existence of row factors in LU decomposition to factors appearing in the Smith normal form of the matrix. For statistical factors, we identify mechanisms and give estimates of the frequency. Our conclusions are tested by experimental data. For pivoting strategies, we compare the sizes of output factors obtained by different strategies. We also comment on timing differences.