Preconditioning complex symmetric linear systems

TL;DR

This paper introduces a novel polynomial preconditioner for complex symmetric linear systems using Hermitian and skew-Hermitian splitting, enhancing iterative solver efficiency without spectrum estimation.

Contribution

It presents a new preconditioning method based on HSS for complex symmetric systems, including an inexact variant, with stability analysis and no need for spectrum estimation.

Findings

Efficient and robust preconditioner demonstrated through numerical results.

Inexact variant reduces computational cost while maintaining effectiveness.

Provides an upper bound for the condition number of the preconditioned system.

Abstract

A new polynomial preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to Conjugate Orthogonal Conjugate Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. Moreover, to reduce the computational cost, an inexact variant based on incomplete Cholesky decomposition or orthogonal polynomials is proposed. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the method completes the description of the preconditioner.