# Reduction of a Regular Matrix Pair (A, B) to Block Hessenberg Triangular Form

@inproceedings{Dackland1995ReductionOA, title={Reduction of a Regular Matrix Pair (A, B) to Block Hessenberg Triangular Form}, author={Krister Dackland and Bo K{\aa}gstr{\"o}m}, booktitle={PARA}, year={1995} }

An algorithm for reduction of a regular matrix pair (A, B) to block Hessenberg-triangular form is presented. This condensed form Q T (A,B)Z = (H,T), where H and T axe block upper Hessenberg and upper triangular, respectively, and Q and Z orthogonal, may serve as a first step in the solution of the generalized eigenvalue problem Ax = λBx. It is shown how an elementwise algorithm can be reorganized in terms of blocked factorizations and higher level BLAS operations. Several ways to annihilate…

## 10 Citations

### Parallel Reduction of a Block Hessenberg-Triangular Matrix Pair to Hessenberg-Triangular Form — Algorithm Design and Performance Results

- Computer Science
- 2004

Performance results for the ScaLAPACK-style implementation show that the parallel algorithm can be used to solve large scale problems effectively.

### Parallel Two-Stage Reduction of a Regular Matrix Pair to Hessenberg-Triangular Form

- Computer SciencePARA
- 2000

A parallel two-stage algorithm for reduction of a regular matrix pair to Hessenberg-triangular form (H, T) is presented and performance results show that the parallel algorithms can be used to solve large scale problems effectively.

### Blocked Algorithms for Reduction of a Regular Matrix Pair to Generalized Schur Form

- Computer SciencePPSC
- 1999

This contribution considers the problem of transforming a regular matrix pair (A;B) to generalized Schur form and presents a parallel version of one of the blocked variants, denoted p blockHT, based on Householder re ections and compact WY representation of the Householder matrices.

### A ScaLAPACK-Style Algorithm for Reducing a Regular Matrix Pair to Block Hessenberg-Triangular Form

- Computer SciencePARA
- 1998

It is shown how a sequential elementwise algorithm can be reorganized in terms of blocked factorizations and matrix-matrix operations to form a parallel algorithm for reduction of a regular matrix pair to block Hessenberg-triangular form.

### Blocked algorithms and software for reduction of a regular matrix pair to generalized Schur form

- Computer ScienceTOMS
- 1999

A two-stage blocked algorithm for reduction of a regular matrix pair (<italic>A , B </italic>) to upper Hessenberg-triangular form is presented and a blocked variant of the single-diagonal double-shift QZ method for computing the generalized Schur form of (<itali>A, B</italic>, which outperforms the current LAPACK routines by a factor 2-5 for sufficiently large problems.

### Contributions to Parallel Algorithms for Sylvester-type Matrix Equations and Periodic Eigenvalue Reordering in Cyclic Matrix Products

- Mathematics, Computer Science
- 2005

A direct method for periodic eigen value reordering in the periodic real Schur form which extends earlier work on the standard and the generalized eigenvalue problems.

### I/O Efficient Algorithms for Matrix Computations

- Computer ScienceArXiv
- 2010

It is shown that techniques like rescheduling of computational steps, appropriate choosing of the blocking parameters and incorporating of more matrix-matrix operations, can be used to improve the I/O and seek complexities of matrix computations.

### A Hierarchical Approach for Performance Analysis of ScaLAPACK-Based Routines Using the Distributed Linear Algebra Machine

- Computer SciencePARA
- 1996

An hierarchical approach for design of performance models for parallel algorithms in linear algebra based on a parallel machine model and the hierarchical structure of the ScaLAPACK library is presented.

### Distributed One-Stage Hessenberg-Triangular Reduction with Wavefront Scheduling

- Computer ScienceSIAM J. Sci. Comput.
- 2018

A novel parallel formulation of Hessenberg-triangular reduction of a regular matrix pair on distributed memory computers is presented, based on a sequential cache-blocked algorit ...

### Parallel Algorithms and Library Software for the Generalized Eigenvalue Problem on Distributed Memory Computer Systems

- Computer Science
- 2016

We present and discuss algorithms and library software for solving the generalized non-symmetric eigenvalue problem (GNEP) on high performance computing (HPC) platforms with distributed memory. Suc…

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