Algorithms for Hessenberg-Triangular Reduction of Fiedler Linearization of Matrix Polynomials

Smallto medium-sized polynomial eigenvalue problems can be solved by linearizing the matrix polynomial and solving the resulting generalized eigenvalue problem using the QZ algorithm. The QZ algorithm, in turn, requires an initial reduction of a matrix pair to Hessenberg– triangular form. In this paper, we discuss the design and evaluation of high-performance parallel algorithms and software for Hessenberg–triangular reduction of a specific linearization of matrix polynomials of arbitrary degree. The proposed algorithm exploits the sparsity structure of the linearization to reduce the number of operations and improve the cache reuse compared to existing algorithms for unstructured inputs. Experiments on both a workstation and an HPC system demonstrate that our structure-exploiting parallel implementation can outperform both the general LAPACK routine DGGHRD and the prototype implementation DGGHR3 of a general blocked algorithm.