Symmetric Permutations for I‐Matrices to Avoid Small Pivots During Incomplete Factorization

Abstract

AbstractIncomplete LU–factorizations have been very successful as preconditioners for solving sparse linear systems iteratively. However, for unsymmetric, indefinite systems small pivots (or even zero pivots) are often very detrimental to the quality of the preconditioner. A fairly recent strategy to deal with this problem has been to permute the rows of the matrix and to scale rows and columns to produce an I–matrix, a matrix having elements of modulus one on the diagonal and elements of at most modulus one elsewhere. These matrices are generally more suited for incomplete LU–factorization. I–matrices are preserved by symmetric permutation, i.e. by applying the same permutation to rows and columns of a matrix. We discuss different approaches for constructing such permutations which aim at improving the sparsity and diagonal dominance of an initial block. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)