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Robust level-3 BLAS Inverse Iteration from the Hessenberg Matrix
arXiv - CS - Mathematical Software Pub Date : 2021-01-13 , DOI: arxiv-2101.05063
Angelika Schwarz

Inverse iteration is known to be an effective method for computing eigenvectors corresponding to simple and well-separated eigenvalues. In the non-symmetric case, the solution of shifted Hessenberg systems is a central step. Existing inverse iteration solvers approach the solution of the shifted Hessenberg systems with either RQ or LU factorizations and, once factored, solve the corresponding systems. This approach has limited level-3 BLAS potential since distinct shifts have distinct factorizations. This paper rearranges the RQ approach such that data shared between distinct shifts is exposed. Thereby the backward substitution with the triangular R factor can be expressed mostly with matrix-matrix multiplications (level-3 BLAS). The resulting algorithm computes eigenvectors in a tiled, overflow-free, and task-parallel fashion. The numerical experiments show that the new algorithm outperforms existing inverse iteration solvers for the computation of both real and complex eigenvectors.

中文翻译:

Hessenberg矩阵的鲁棒3级BLAS逆迭代

已知逆迭代是一种用于计算与简单且分离良好的特征值相对应的特征向量的有效方法。在非对称情况下,转移的Hessenberg系统的求解是中心步骤。现有的逆迭代求解器通过RQ或LU分解来解决移位后的Hessenberg系统的解,并在分解后求解相应的系统。该方法具有有限的3级BLAS潜力,因为不同的变化具有不同的因式分解。本文对RQ方法进行了重新安排,使不同班次之间共享的数据得以公开。因此,可以用矩阵矩阵乘法(级别3 BLAS)来表示用三角形R因子进行的向后替换。结果算法以分块,无溢出和任务并行的方式计算特征向量。
更新日期:2021-01-14
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