将eigCG算法推广到多右端线性方程组的非对称Lanczos问题
Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple right-hand sides
摘要：
该技术是用来建立稀疏对称线性系统的EigCG算法扩展到非对称的情况下使用的BiCG算法。我们表明，类似于对称的情况下，我们可以建立一个算法，能够计算几个最小幅度的特征值和它们相应的左，右特征向量的非对称矩阵使用只有一个小窗口的BiCG残差，同时解决线性系统与该矩阵。对于多个右手边的系统，我们给出了一个算法，该算法在求解前几个系统的同时计算出更多的特征值，然后使用计算出的特征向量来压缩剩余系统的BiCGStab。我们的实验上的各种测试问题，包括格点量子色动力学，显示出显着的能力EigBiCG计算谱近似的准确性相媲美的unrestarted，非对称Lanczos。此外，我们的增量EigBiCG，然后适当地重新启动和放气BiCGStab为具有多个右侧的系统提供了一种有竞争力的方法。
英文摘要：
The technique that was used to build the EigCG algorithm for sparse symmetric linear systems is extended to the nonsymmetric case using the BiCG algorithm. We show that, similarly to the symmetric case, we can build an algorithm that is capable of computing a few smallest magnitude eigenvalues and their corresponding left and right eigenvectors of a nonsymmetric matrix using only a small window of the BiCG residuals while simultaneously solving a linear system with that matrix. For a system with multiple right-hand sides, we give an algorithm that computes incrementally more eigenvalues while solving the first few systems and then uses the computed eigenvectors to deflate BiCGStab for the remaining systems. Our experiments on various test problems, including Lattice QCD, show the remarkable ability of EigBiCG to compute spectral approximations with accuracy comparable to that of the unrestarted, nonsymmetric Lanczos. Furthermore, our incremental EigBiCG followed by appropriately restarted and deflated BiCGStab provides a competitive method for systems with multiple right-hand sides.
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