Common Factors in Fraction-Free Matrix Decompositions

Johannes Middeke,David J Jeffrey,Christoph Koutschan

doi:10.1007/s11786-020-00495-9

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.

Highlights

Known earlier to Dodgson [8] and Jordan1, the fraction-free method for exact matrix computations became well known because of its application by Bareiss [1] to the solution of a linear system over Z, and later over an integral domain [2]
In linear algebra, equation solving is related to the matrix factorizations LU and Q R, it is natural that fraction-free methods would be extended later to those factorizations
Fraction-free Gram–Schmidt orthogonalization and Q R factorization were studied in Erlingsson et al [10] and Zhou and Jeffrey [26]

Summary

Introduction

Known earlier to Dodgson [8] and Jordan (see Durand [9]), the fraction-free method for exact matrix computations became well known because of its application by Bareiss [1] to the solution of a linear system over Z, and later over an integral domain [2]. The key feature of Bareiss’s algorithm is that it creates factors which are common to every element in a row, but which can be removed by exact divisions. For a matrix A ∈ Dm×n of rank r we say that A = Pr L D−1U Pc is a fraction-free LU decomposition if Pr ∈ Dm×m and Pc ∈ Dn×n are permutation matrices, L ∈ Dm×r has Li j = 0 for j > i and Lii = 0 for all i , U ∈ Dr×n has Ui j = 0 for i > j and Uii = 0 for all i , and D ∈ Dr×r is a diagonal matrix (with full rank). The result asserts that we can cancel all common factors which we find from the final output This yields a fraction-free LU decomposition of A where the size of the entries of U (and L) are smaller than in the L D−1U decomposition. It will be an interesting discussion for future research whether it is better to cancel as many factors as possible from U or to cancel them from L

LU and the Smith–Jacobson Normal Form

Efficient Detection of Factors

Expected Number of Factors

Q R Decomposition

Findings

A11 A12 A21 A22