Robust level-3 BLAS Inverse Iteration from the Hessenberg Matrix

TL;DR

This paper introduces a novel level-3 BLAS optimized inverse iteration method for Hessenberg matrices that improves efficiency and parallelism in computing eigenvectors, outperforming existing solvers.

Contribution

It presents a rearranged RQ-based inverse iteration algorithm that enhances data sharing and leverages level-3 BLAS for better performance.

Findings

Outperforms existing inverse iteration solvers in speed.

Effective for both real and complex eigenvector computations.

Provides a tiled, overflow-free, task-parallel implementation.

Abstract

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.