将eigCG算法推广到多右端非对称线性方程组
Extending the eigCG algorithm to non-symmetric linear systems with multiple right-hand sides
摘要：
对于Hermitian正定线性系统和特征值问题，eigCG算法是一种内存有效的算法，它可以求解线性系统并同时计算其一些特征值。该算法是基于共轭梯度算法（CG），但它只使用一个窗口的CG算法产生的向量计算近似特征值。特征向量的数量和精度可以通过求解更多的右侧来增加。对于具有多个右端的厄米特系统，计算出的特征向量可以用来加速后续系统的求解。通过求解正则方程，在格点量子色动力学问题上对该算法进行了测试，结果表明，当我们接近轻夸克质量时，该算法具有很大的加速因子，并消除了临界慢化。在这里，扩展到非对称的情况下，双边Lanczos算法的基础上。新算法在格点量子色动力学问题上得到了较好的结果。我们还研究了临界减速的去除，并与eigCG算法的结果进行了比较。我们还讨论了系统是gamma 5-Hermitian的情况。
英文摘要：
For Hermitian positive definite linear systems and eigenvalue problems, the eigCG algorithm is a memory efficient algorithm that solves the linear system and simultaneously computes some of its eigenvalues. The algorithm is based on the Conjugate-Gradient (CG) algorithm, however, it uses only a window of the vectors generated by the CG algorithm to compute approximate eigenvalues. The number and accuracy of the eigenvectors can be increased by solving more right-hand sides. For Hermitian systems with multiple right-hand sides, the computed eigenvectors can be used to speed up the solution of subsequent systems. The algorithm was tested on Lattice QCD problems by solving the normal equations and was shown to give large speed up factors and to remove the critical slowing down as we approach light quark masses. Here, an extension to the non-symmetric case based on the two-sided Lanczos algorithm is given. The new algorithm is tested on Lattice QCD problems and is shown to give promising results. We also study the removal of the critical slowing down and compare results with those of the eigCG algorithm. We also discuss the case when the system is gamma5-Hermitian.
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