On a weighted quasi-residual minimization strategy of the QMR method for solving complex symmetric shifted linear systems

TL;DR

This paper introduces a weighted quasi-residual minimization strategy for the QMR method to efficiently solve complex symmetric shifted linear systems, which are common in large-scale electronic structure simulations.

Contribution

It proposes a new weighted QMR variant, called shifted QMR_SYM(B), optimized for complex symmetric shifted systems, enhancing computational efficiency.

Findings

The shifted QMR_SYM(B) method demonstrates improved performance in numerical experiments.

The proposed approach reduces computational costs compared to traditional methods.

Numerical examples confirm the effectiveness of the weighted strategy.

Abstract

We consider the solution of complex symmetric shifted linear systems. Such systems arise in large-scale electronic structure simulations and there is a strong need for the fast solution of the systems. With the aim of solving the systems efficiently, we consider a special case of the QMR method for non-Hermitian shifted linear systems and propose its weighted quasi-minimal residual approach. A numerical algorithm, referred to as shifted QMR\_SYM($B$), is given by the choice of a particularly cost-effective weight. Numerical examples are presented to show the performance of the shifted QMR\_SYM($B$) method.