Matrix enlarging methods and their application

Abstract

This paper explores several methods for matrix enlarging, where an enlarged matrix à is constructed from a given matrix A. The methods explored include matrix primitization, stretching and node splitting. Graph interpretations of these methods are provided. Solving linear problems using enlarged matrices yields the answer to the original Ax = b problem. à can exhibit several desirable properties. For example, à can be constructed so that the valence of any row and/or column is smaller than some desired number (≥ 4). This is beneficial for algorithms that depend on the square of the number of entries of a row or column. Most particularly, matrix enlarging can result in a reduction of the fill-in in the R matrix which occurs during orthogonal factorization as a result of dense rows. Numerical experiments support these conjectures.