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.

中文翻译：

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TriCG 和 TriMR：对称拟定系统的两种迭代方法

我们基于 Saunders、Simon 和 Yip 在 1988 年提出的正交三对角化过程，引入了名为 TriCG 和 TriMR 的迭代方法来求解对称拟定系统。 TriCG 和 TriMR 相当于预处理块 CG 和块 MINRES，具有两个右手边在每次迭代时对两个近似解求和的边，但每次迭代需要更少的存储和工作。我们评估了 TriCG 和 TriMR 在从 SuiteSparse 矩阵集合和离散化和稳定的斯托克斯方程生成的线性系统上的性能。我们将 TriCG 和 TriMR 与 SYMMLQ 和 MINRES（推荐用于对称和不定系统的 Krylov 方法）进行比较。在我们所有的实验中，