A novel, blocked algorithm for the reduction to Hessenberg-triangular form

Abstract

We present an alternative algorithm and implementation for theHessenberg-triangular reduction, an essential step in the QZalgorithm for solving generalized eigenvalue problems. Thereduction step has a cubic computational complexity, and hence,high-performance implementations are compulsory for keeping thecomputing time under control. Our algorithm is of simplemathematical nature and relies on the connection betweengeneralized and classical eigenvalue problems. Via system solving andthe classical reduction of a single matrix to Hessenberg form, we areable to get a theoretically equivalent reduction toHessenberg-triangular form. As a result, we can perform most of thecomputational work by relying on existing, highly efficient implementations,which make extensive use of blocking. The accompanying error analysisshows that preprocessing and iterative refinement can benecessary to achieve accurate results. Numerical results showcompetitiveness with existing implementations.