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.

翻译：我们采用称为TriCG和TriMR的迭代方法,根据Saunders、Simon和Yip于1988年提出的正方形三对角化进程解决对称准定界系系。TriCG和TriMR相当于Clock-CG和Block-MINRES的前提条件,有两个右侧,其中两种近似解决办法在每次迭代中都得到总和,但每迭代中需要的储存和工作较少。我们评估TriCG和TriMR在从SuiteSparse 矩阵采集以及离散和稳定式斯托克斯方程式生成的线性系统中的性能。我们将TriCG和TriMR与SyMLQ和MINRES进行比较。推荐的Krylov和TriMRMR方法,在我们的实验中,TriCG和TriMRMR早于SIMLQ和MINRES的剩余停止状态,在迭代数方面改进到50%。它们也比Bl-C和BMINCR-C在三-C-C-G-CRMIT-C-G-C-C-C-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-C-C-C-C-Q-C-C-C-C-C-C-C-Q-Q-G-G-G-G-G-G-G-G-G-G-G-Q-Q-Q-Q-Q-C-C-C-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-C-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-T-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-Q-T-T-