基于Hessenberg矩阵的鲁棒三级BLAS逆迭代
Robust level-3 BLAS Inverse Iteration from the Hessenberg Matrix
摘要：
众所周知，逆迭代是计算简单且分离良好的特征值对应的特征向量的有效方法。在非对称情况下，位移Hessenberg系统的解是一个中心步骤。现有的逆迭代求解器的解决方案的移位Hessenberg系统的RQ或LU分解，一旦分解，解决相应的系统。这种方法具有有限的3级BLAS潜力，因为不同的移位具有不同的因子分解。本文重新安排的RQ方法，使不同的班次之间共享的数据是公开的。因此，三角R因子的向后替换主要可以用矩阵-矩阵乘法（3级BLAS）来表示。由此产生的算法计算特征向量的平铺，无溢出，任务并行的方式。数值实验表明，新算法在计算实特征向量和复特征向量时均优于现有的逆迭代求解器。
英文摘要：
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.
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