Differential qd algorithm with shifts for rank-structured matrices

TL;DR

This paper introduces variants of the differential qd algorithm tailored for rank-structured matrices, including a generalization of dqds for Hessenberg matrices, enhancing eigenvalue computations for specific matrix classes.

Contribution

The paper develops new qd algorithm variants for quasiseparable matrices, including a direct generalization of dqds for Hessenberg matrices, applicable to polynomial root finding and eigenvalue problems.

Findings

Preliminary numerical experiments show promising results.

The generalized dqds algorithm preserves high relative accuracy.

Applicable to matrices like companion and confederate for polynomial roots.

Abstract

Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable. In particular, in the symmetric tridiagonal case where differential qd algorithm with shifts (dqds) proposed by Fernando and Parlett enjoys often faster convergence while preserving high relative accuracy (that is not guaranteed in QR algorithm). In eigenvalue computations for rank-structured matrices QR algorithm is also a popular choice since, in the symmetric case, the rank structure is preserved. In the unsymmetric case, however, QR algorithm destroys the rank structure and, hence, LR-type algorithms come to play once again. In the current paper we discover several variants of qd algorithms for quasiseparable matrices. Remarkably, one of them, when applied to Hessenberg matrices becomes a direct generalization of dqds algorithm for tridiagonal matrices. Therefore, it can be applied to such important matrices as companion and confederate, and provides an alternative algorithm for finding roots of a polynomial represented in the basis of orthogonal polynomials. Results of preliminary numerical experiments are presented.