The RQR algorithm

TL;DR

The paper addresses solving the standard eigenvalue problem for a single matrix $A$ by extending pole-swapping, previously used for generalized eigenproblems, to the pencil $A - \\lambda I$. The method, termed RQR, preserves a unitary Hessenberg factor $U$ stored as core transformations $U = U_1 U_2 \\cdots U_{n-1}$ and uses type I and II moves with shifts to move and swap poles, aiming for rapid convergence with deflation reducing problem size. Numerical experiments show that RQR is faster and more accurate than the QR-based bulge-chasing approach on several test matrices, indicating that pole-swapping is a viable alternative for standard eigenproblems. The paper discusses potential enhancements (multishift, aggressive deflation) and concludes that the current results already demonstrate substantial promise for RQR in competitive eigenvalue computations.

Abstract

Pole-swapping algorithms, generalizations of bulge-chasing algorithms, have been shown to be a viable alternative to the bulge-chasing QZ algorithm for solving the generalized eigenvalue problem for a matrix pencil A - λB. It is natural to try to devise a pole-swapping algorithm that solves the standard eigenvalue problem for a single matrix A. This paper introduces such an algorithm and shows that it is competitive with Francis's bulge-chasing QR algorithm.

Figures (1)

• Figure 5.1: Comparison of the RQR pole swapping algorithm (in red) with the QR bulge chasing algorithm (in blue) in terms of backward error (left) and runtime (right) for random matrices reduced to upper Hessenberg form (square markers) and $i + j$ upper Hessenberg matrices (dashed line).