Paper Info
Title
A Householder-Based Algorithm for Hessenberg-Triangular Reduction
Abstract
The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil A - lambda B requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using matrix multiplication routines. A nonvanishing fraction of the total flop-count must nevertheless still be performed as sequences of overlapping Givens rotations alternately applied from the left and from the right. The many data dependencies associated with this computational pattern leads to inefficient use of the processor and poor scalability. In this paper, we therefore introduce a fundamentally different approach that relies entirely on (large) Householder reflectors partially accumulated into block reflectors, by using (compact) WY representations. Even though the new algorithm requires more floating point operations than the state-of-the-art algorithm, extensive experiments on both real and synthetic data indicate that it is still competitive, even in a sequential setting. The new algorithm is conjectured to have better parallel scalability, an idea which is partially supported by early small-scale experiments using multithreaded BLAS. The design and evaluation of a parallel formulation is future work.
Year
DOI
Venue
2018
10.1137/17M1153637
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Keywords
Field
DocType
Hessenberg-triangular reduction,Householder reflectors,iterative refinement
Matrix pencil,FLOPS,Floating point,Matrix (mathematics),Algorithm,Synthetic data,Matrix multiplication,Eigenvalues and eigenvectors,Mathematics,Scalability
Journal
Volume
Issue
ISSN
39
3
0895-4798
Citations 
PageRank 
References 
0
0.34
11
Authors
3
Name
Order
Citations
PageRank
Zvonimir Bujanović1123.14
Lars Karlsson2515.16
Daniel Kressner344948.01