Direct solution of near-symmetric matrices and its applications

Abstract

By a near-symmetric matrix, we mean that only a very few of entries in the matrix are non-symmetric. If those non-symmetric entries above the diagonal are replaced with the corresponding entries below the diagonal, it will become symmetric. Such a matrix can be encountered in the analysis of nonlinear continuum problems. Based on Sherman-Morrison's formula, a new scheme for decomposing near-symmetric matrices is proposed, which is much more effective and less memory-used than those solvers for common sparse non-symmetric matrices under the condition that the numerical stability is assured. Moreover, the solver corresponding to the scheme, which is suitable for both symmetric and non-symmetric matrices, can be developed through slightly augmenting the solvers based on LDLT decomposition. With an example on a frictional contact problem, the advantages of the proposed scheme are illustrated.