Global Convergence of Hessenberg Shifted QR III: Approximate Ritz Values via Shifted Inverse Iteration

TL;DR

This paper introduces a randomized shifted inverse iteration algorithm that accurately computes eigenvalues of small matrices with provable error bounds, potentially aiding in Ritz value computations within shifted QR methods.

Contribution

The paper presents a novel, self-contained randomized algorithm for eigenvalue computation with rigorous backward error guarantees, applicable to small matrices and useful for Ritz value calculations.

Findings

Algorithm computes eigenvalues up to backward error in polynomial time.

Complexity is high for large matrices but feasible for small matrices.

Potential application in shifted QR algorithms for eigenvalue problems.

Abstract

We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\|M\|$ in $O(n^4+n^3\log^2(n/\delta)+\log(n/\delta)^2\log\log(n/\delta))$ floating point operations using $O(\log^2(n/\delta))$ bits of precision. While the $O(n^4)$ complexity is prohibitive for large matrices, the algorithm is simple and may be useful for provably computing the eigenvalues of small matrices using controlled precision, in particular for computing Ritz values in shifted QR algorithms as in (Banks, Garza-Vargas, Srivastava, 2022).