A New Real Structure-preserving Quaternion QR Algorithm

TL;DR

This paper introduces new structure-preserving quaternion QR algorithms that are fast, robust, and accurate for eigenproblems of quaternion matrices, utilizing orthogonally JRS-symplectic transformations and novel quaternion Givens matrices.

Contribution

It develops the first JRS-QR algorithms for JRS-symmetric matrices and introduces a quaternion Givens matrix for Hessenberg matrices, advancing quaternion eigenproblem solutions.

Findings

Algorithms demonstrate high efficiency in numerical experiments.

Methods achieve accurate eigenvalue computations.

New techniques simplify quaternion matrix decompositions.

Abstract

New real structure-preserving decompositions are introduced to develop fast and robust algorithms for the (right) eigenproblem of general quaternion matrices. Under the orthogonally JRS-symplectic transformations, the Francis JRS-QR step and the JRS-QR algorithm are firstly proposed for JRS-symmetric matrices and then applied to calculate the Schur forms of quaternion matrices. A novel quaternion Givens matrix is defined and utilized to compute the QR factorization of quaternion Hessenberg matrices. An implicit double shift quaternion QR algorithm is presented with a technique for automatically choosing shifts and within real operations. Numerical experiments are provided to demonstrate the efficiency and accuracy of newly proposed algorithms.