Improved TriCG and TriMR Methods for Symmetric Quasi‐Definite Linear Systems
Kui Du, Jiajun Fan, Y. Zhang · Numerical Linear Algebra with Applications · 2025
ABSTRACT TriCG and TriMR are two short‐recurrence iterative methods proposed in recent years for solving symmetric quasi‐definite (SQD) linear systems. The basic mechanism underlying TriCG and TriMR is the generalized Saunders–Simon–Yip tridiagonalization process (gSSY). In this work, we show that TriCG and TriMR may fail to find the solution of SQD linear systems when gSSY terminates. To avoid this issue, we propose an improved generalized Saunders–Simon–Yip tridiagonalization process (igSSY). Based on igSSY, we introduce two new iterative methods named iTriCG and iTriMR for solving SQD linear systems in the same fashion as TriCG and TriMR. We show that iTriCG and iTriMR always terminate with the solution of SQD linear systems when igSSY terminates. In addition, we prove that the upper triangular factor of the QR factorization used in TriMR has only three nonzero diagonals, and based on this fact, we provide simplified short recurrences for TriMR, which reduce the work per iteration. These simplified short recurrences also apply to iTriMR. Numerical experiments are given to illustrate the merits of iTriCG and iTriMR. MSC2020 Classification: 15A06, 65F10, 65F25, 65F50