We introduce two iterative methods, GPBiLQ and GPQMR, for solving unsymmetric partitioned linear systems. The basic mechanism underlying GPBiLQ and GPQMR is a novel simultaneous tridiagonalization via biorthogonality that allows for short-recurrence iterative schemes. Similar to the biconjugate gradient method, it is possible to develop another method, GPBiCG, whose iterate (if it exists) can be obtained inexpensively from the GPBiLQ iterate. Whereas the iterate of GPBiCG may not exist, the iterates of GPBiLQ and GPQMR are always well defined as long as the biorthogonal tridiagonal reduction process does not break down. We discuss connections between the proposed methods and some existing methods, and give numerical experiments to illustrate the performance of the proposed methods.

翻译：本文提出了两种迭代方法——GPBiLQ和GPQMR，用于求解非对称分块线性系统。GPBiLQ和GPQMR的基本机制是通过双正交性实现的同时三对角化过程，从而允许短递推迭代方案。类似于双共轭梯度法，可进一步开发另一种方法GPBiCG，其迭代解（若存在）可通过GPBiLQ迭代解廉价计算得到。尽管GPBiCG的迭代解可能不存在，但只要双正交三对角约化过程不发生中断，GPBiLQ和GPQMR的迭代解始终定义良好。我们讨论了所提方法与现有方法之间的联系，并通过数值实验验证了所提方法的性能。