High quality preconditioning of a general symmetric positive definite matrix based on its UTU + UTR + RTU‐decomposition

Abstract

A new matrix decomposition of the form A = UTU + UTR + RTU is proposed and investigated, where U is an upper triangular matrix (an approximation to the exact Cholesky factor U0), and R is a strictly upper triangular error matrix (with small elements and the fill-in limited by that of U0). For an arbitrary symmetric positive matrix A such a decomposition always exists and can be efficiently constructed; however it is not unique, and is determined by the choice of an involved truncation rule. An analysis of both spectral and K-condition numbers is given for the preconditioned matrix M = U−T AU−1 and a comparison is made with the RIC preconditioning proposed by Ajiz and Jennings. A concept of approximation order of an incomplete factorization is introduced and it is shown that RIC is the first order method, whereas the proposed method is of second order. The idea underlying the proposed method is also applicable to the analysis of CGNE-type methods for general non-singular matrices and approximate LU factorizations of non-symmetric positive definite matrices. Practical use of the preconditioning techniques developed is discussed and illustrated by an extensive set of numerical examples. Copyright © 1999 John Wiley & Sons, Ltd.