GPBiLQ and GPQMR: Two iterative methods for unsymmetric partitioned linear systems

TL;DR

This paper introduces two new iterative methods, GPBiLQ and GPQMR, for efficiently solving unsymmetric partitioned linear systems using a novel biorthogonal tridiagonalization approach, with demonstrated numerical effectiveness.

Contribution

The paper presents two novel iterative algorithms, GPBiLQ and GPQMR, based on a new simultaneous biorthogonal tridiagonalization technique for unsymmetric linear systems.

Findings

GPBiLQ and GPQMR are always well-defined under certain conditions.

Numerical experiments show competitive performance of the proposed methods.

Connections to existing methods are discussed and analyzed.

Abstract

We introduce two iterative methods, GPBiLQ and GPQMR, for solving unsymmetric partitioned linear systems. The basic mechanism underlying GPBiLQ and GPQMR is a novel simultaneous tridiagonalization via biorthogonality that allows for short-recurrence iterative schemes. Similar to the biconjugate gradient method, it is possible to develop another method, GPBiCG, whose iterate (if it exists) can be obtained inexpensively from the GPBiLQ iterate. Whereas the iterate of GPBiCG may not exist, the iterates of GPBiLQ and GPQMR are always well defined as long as the biorthogonal tridiagonal reduction process does not break down. We discuss connections between the proposed methods and some existing methods, and give numerical experiments to illustrate the performance of the proposed methods.