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We introduce iterative methods named TriCG and TriMR for solving symmetricquasi-definite systems based on the orthogonal tridiagonalization processproposed by Saunders, Simon and Yip in 1988. TriCG and TriMR are tantamount topreconditioned Block-CG and Block-MINRES with two right-hand sides in which thetwo approximate solutions are summed at each iteration, but require lessstorage and work per iteration. We evaluate the performance of TriCG and TriMRon linear systems generated from the SuiteSparse Matrix Collection and fromdiscretized and stablized Stokes equations. We compare TriCG and TriMR withSYMMLQ and MINRES, the recommended Krylov methods for symmetric and indefinitesystems. In all our experiments, TriCG and TriMR terminate earlier than SYMMLQand MINRES on a residual-based stopping condition with an improvement of up to50% in terms of number of iterations. They also terminate more reliably thanBlock-CG and Block-MINRES. Experiments in quadruple and octuple precisionsuggest that loss of orthogonality in the basis vectors is significantly lesspronounced in TriCG and TriMR than in Block-CG and Block-MINRES.

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