Common Factors in Fraction-Free Matrix Decompositions

TL;DR

This paper investigates the presence of common row factors in fraction-free LU and QR matrix decompositions, distinguishing between systematic and statistical factors, and relates these to matrix properties and data characteristics.

Contribution

It identifies and characterizes systematic and statistical common factors in fraction-free decompositions, linking them to matrix invariants and data-dependent mechanisms.

Findings

Fraction-free LU decompositions often have common row factors.

Systematic factors are independent of data and relate to the reduction process.

Statistical factors depend on data and occur with measurable frequency.

Abstract

We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauss--Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith--Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally.