TriCG and TriMR: Two Iterative Methods for Symmetric Quasi-Definite Systems

TL;DR

This paper introduces TriCG and TriMR, new iterative methods for symmetric quasi-definite systems that outperform existing methods in efficiency and reliability, especially in high-precision computations.

Contribution

The paper presents two novel iterative algorithms, TriCG and TriMR, based on orthogonal tridiagonalization, offering improved performance over traditional Krylov methods for symmetric quasi-definite systems.

Findings

TriCG and TriMR terminate earlier than SYMMLQ and MINRES with up to 50% fewer iterations.

They are more reliable than Block-CG and Block-MINRES in convergence.

Loss of orthogonality is significantly less in TriCG and TriMR at high precision.

Abstract

We introduce iterative methods named TriCG and TriMR for solving symmetric quasi-definite systems based on the orthogonal tridiagonalization process proposed by Saunders, Simon and Yip in 1988. TriCG and TriMR are tantamount to preconditioned Block-CG and Block-MINRES with two right-hand sides in which the two approximate solutions are summed at each iteration, but require less storage and work per iteration. We evaluate the performance of TriCG and TriMR on linear systems generated from the SuiteSparse Matrix Collection and from discretized and stablized Stokes equations. We compare TriCG and TriMR with SYMMLQ and MINRES, the recommended Krylov methods for symmetric and indefinite systems. In all our experiments, TriCG and TriMR terminate earlier than SYMMLQ and MINRES on a residual-based stopping condition with an improvement of up to 50% in terms of number of iterations. They also terminate more reliably than Block-CG and Block-MINRES. Experiments in quadruple and octuple precision suggest that loss of orthogonality in the basis vectors is significantly less pronounced in TriCG and TriMR than in Block-CG and Block-MINRES.