research-article

Efficient generalized Hessenberg form and applications

Authors:

Zvonimir Bujanović,

Zlatko DrmačAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 39, Issue 3

Article No.: 19, Pages 1 - 19

https://doi.org/10.1145/2450153.2450157

Published: 03 May 2013 Publication History

Abstract

This article proposes an efficient algorithm for reducing matrices to generalized Hessenberg form by unitary similarity, and recommends using it as a preprocessor in a variety of applications. To illustrate its usefulness, two cases from control theory are analyzed in detail: a solution procedure for a sequence of shifted linear systems with multiple right hand sides (e.g. evaluating the transfer function of a MIMO LTI dynamical system at many points) and computation of the staircase form. The proposed algorithm for the generalized Hessenberg reduction uses two levels of aggregation of Householder reflectors, thus allowing efficient BLAS 3-based computation. Another level of aggregation is introduced when solving many shifted systems by processing the shifts in batches. Numerical experiments confirm that the proposed methods have superior efficiency.

References

[1]

Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Croz, J. D., Greenbaum, A., Hammarling, S., McKenny, A., Ostrouchov, S., and Sorensen, D. 1992. LAPACK Users' Guide 2nd Ed. SIAM, Philadelphia, PA.

Digital Library

[2]

Beattie, C., Drmač, Z., and Gugercin, S. 2011. A note on shifted Hessenberg systems and frequency response computation. ACM Trans. Math. Softw. 38, 2.

Digital Library

[3]

Bischof, C., Lang, B., and Sun, X. 2000. A framework for symmetric band reduction. ACM Trans. Math. Softw. 26, 4, 581--601.

Digital Library

[4]

Bischof, C. and Tang, P. T. P. 1991a. Generalized incremental condition estimation. Tech. rep. 32, LAPACK working note. http://www.netlib.org/lapack/lawnspdf/lawn32.pdf.

Digital Library

[5]

Bischof, C. and Tang, P. T. P. 1991b. Robust incremental condition estimation. Tech. rep. 33, LAPACK working note. http://www.netlib.org/lapack/lawnspdf/lawn33.pdf.

Digital Library

[6]

Bischof, C. and van Loan, C. F. 1987. The WY representation for products of Householder matrices. SIAM J. Sci. Stat. Comput. 8, 1, 2--13.

Digital Library

[7]

Blackford, L. S., Choi J., Cleary, A., D'Azeuedo, E., Demmel, J., Dhillon, I., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., and Whaley, R. C. 1997. ScaLAPACK User's Guide. SIAM, Philadelphia, PA.

Digital Library

[8]

Bosner, N. 2011. Algorithms for simultaneous reduction to the m-Hessenberg--triangular--triangular form. Tech. rep., University of Zagreb.

[9]

Bujanović, Z. 2011. Krylov type methods for large scale eigenvalue computations. Ph.D. dissertation. Department of Mathematics, University of Zagreb, Croatia.

[10]

Bujanović, Z. and Drmač, Z. 2010. How a numerical rank revealing instability affects computer aided control system design. Tech. rep. 2010-1, SLICOT working note. http://www.slicot.org/REPORTS/SLWN2010-1.pdf.

[11]

Businger, P. A. and Golub, G. H. 1965. Linear least squares solutions by Householder transformations. Numer. Math. 7, 269--276.

Digital Library

[12]

Datta, B. 2004. Numerical Methods for Linear Control Systems. Elsevier Academic Press, Amsterdam.

[13]

Dongarra, J. J., Du Croz, J. J., Duff, I., and Hammarling, S. 1990. A set of Level 3 basic linear algebra subprograms. ACM Trans. Math. Softw. 16, 1, 1--17.

Digital Library

[14]

Drmač, Z. and Bujanović, Z. 2008. On the failure of rank revealing QR factorization software—a case study. ACM Trans. Math. Softw. 35, 2, 1--28.

Digital Library

[15]

Haidar, A., Ltaief, H., and Dongarra, J. 2011. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (SC11). Article 8.

Digital Library

[16]

Henry, G. 1994. The shifted Hessenberg system solve computation. Tech. rep., Cornell Theory Center, Advanced Computing Research Institute.

Digital Library

[17]

Karlsson, L. 2011. Scheduling of parallel matrix computations and data layout conversion for HPC and multi-core architectures. Ph.D. dissertation, Department of Computing Science, Umeå University, Sweden.

[18]

Lehoucq, R. 1997. Truncated QR algorithms and the numerical solution of large scale eigenvalue problems. Tech. rep., Argonne National Laboratory, Argonne IL.

[19]

Mohanty, S. K. 2009. I/O efficient algorithms for matrix computations. Ph.D. dissertation, Department of Computer Science and Engineering, Indian Institute of Technology Guwahati, India.

[20]

Schreiber, R. and van Loan, C. F. 1989. A storage--efficient WY representation for products of Householder transformations. SIAM J. Sci. Stat. Comput. 10, 1, 53--57.

Digital Library

[21]

SLICOT. 2010. The control and systems library. http://www.slicot.org/.

[22]

Tomov, S. and Dongarra, J. 2009. Accelerating the reduction to upper Hessenberg form through hybrid GPU-based computing. Tech. rep. 219, LAPACK working note. http://www.netlib.org/lapack/lawnspdf/lawn219.pdf.

[23]

Tomov, S., Nath, R., and Dongarra, J. 2010. Accelerating the reduction to upper Hessenberg, tridiagonal, and bidiagonal forms through hybrid GPU-based computing. Parallel Comput. 36, 12, 645--654.

Digital Library

[24]

Van Dooren, P. 1979. The computation of Kronecker's canonical form of a singular pencil. Linear Algebra Appl. 27, 122--135.

[25]

Van Dooren, P. 1984. Reduced order observers: A new algorithm and proof. Syst. Control Lett. 4, 243--251.

[26]

Van Loan, C. F. 1982. Using the Hessenberg decomposition in control theory. Math. Program. Study 18, 102--111.

Cited By

Schwarz A(2022)Robust level-3 BLAS Inverse Iteration from the Hessenberg MatrixACM Transactions on Mathematical Software10.1145/354478948:3(1-30)Online publication date: 15-Jul-2022
https://dl.acm.org/doi/10.1145/3544789
Bosner NBujanović ZDrmač Z(2018)Parallel Solver for Shifted Systems in a Hybrid CPU--GPU FrameworkSIAM Journal on Scientific Computing10.1137/17M114446540:4(C605-C633)Online publication date: 19-Jul-2018
https://doi.org/10.1137/17M1144465
Bosner NKarlsson L(2017)Parallel and Heterogeneous $m$--Hessenberg--Triangular--Triangular ReductionSIAM Journal on Scientific Computing10.1137/15M104734939:1(C29-C47)Online publication date: 12-Jan-2017
https://doi.org/10.1137/15M1047349
Show More Cited By

Index Terms

Efficient generalized Hessenberg form and applications
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Linear algebra algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices
  2. Mathematical software

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cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 39, Issue 3

April 2013

149 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/2450153

Issue’s Table of Contents

Copyright © 2013 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 03 May 2013

Accepted: 01 December 2012

Revised: 01 April 2012

Received: 01 November 2011

Published in TOMS Volume 39, Issue 3

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Ministarstvo Obrazovanja, Znanosti i Sporta

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Cited By

Schwarz A(2022)Robust level-3 BLAS Inverse Iteration from the Hessenberg MatrixACM Transactions on Mathematical Software10.1145/354478948:3(1-30)Online publication date: 15-Jul-2022
https://dl.acm.org/doi/10.1145/3544789
Bosner NBujanović ZDrmač Z(2018)Parallel Solver for Shifted Systems in a Hybrid CPU--GPU FrameworkSIAM Journal on Scientific Computing10.1137/17M114446540:4(C605-C633)Online publication date: 19-Jul-2018
https://doi.org/10.1137/17M1144465
Bosner NKarlsson L(2017)Parallel and Heterogeneous $m$--Hessenberg--Triangular--Triangular ReductionSIAM Journal on Scientific Computing10.1137/15M104734939:1(C29-C47)Online publication date: 12-Jan-2017
https://doi.org/10.1137/15M1047349
Bosner N(2015)Efficient algorithm for simultaneous reduction to the $$m$$-Hessenberg-triangular-triangular formBIT10.1007/s10543-014-0516-y55:3(677-703)Online publication date: 1-Sep-2015
https://dl.acm.org/doi/10.1007/s10543-014-0516-y

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