Nela Bosner
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.
- 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. Google Scholar
Digital Library
- Beattie, C., Drmač, Z., and Gugercin, S. 2011. A note on shifted Hessenberg systems and frequency response computation. ACM Trans. Math. Softw. 38, 2. Google ScholarDigital Library
- Bischof, C., Lang, B., and Sun, X. 2000. A framework for symmetric band reduction. ACM Trans. Math. Softw. 26, 4, 581--601. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Bosner, N. 2011. Algorithms for simultaneous reduction to the m-Hessenberg--triangular--triangular form. Tech. rep., University of Zagreb.Google Scholar
- Bujanović, Z. 2011. Krylov type methods for large scale eigenvalue computations. Ph.D. dissertation. Department of Mathematics, University of Zagreb, Croatia.Google Scholar
- 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.Google Scholar
- Businger, P. A. and Golub, G. H. 1965. Linear least squares solutions by Householder transformations. Numer. Math. 7, 269--276.Google ScholarDigital Library
- Datta, B. 2004. Numerical Methods for Linear Control Systems. Elsevier Academic Press, Amsterdam.Google Scholar
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- Henry, G. 1994. The shifted Hessenberg system solve computation. Tech. rep., Cornell Theory Center, Advanced Computing Research Institute. Google ScholarDigital Library
- 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.Google Scholar
- Lehoucq, R. 1997. Truncated QR algorithms and the numerical solution of large scale eigenvalue problems. Tech. rep., Argonne National Laboratory, Argonne IL.Google Scholar
- 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.Google Scholar
- 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. Google ScholarDigital Library
- SLICOT. 2010. The control and systems library. http://www.slicot.org/.Google Scholar
- 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.Google Scholar
- 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. Google ScholarDigital Library
- Van Dooren, P. 1979. The computation of Kronecker's canonical form of a singular pencil. Linear Algebra Appl. 27, 122--135.Google ScholarCross Ref
- Van Dooren, P. 1984. Reduced order observers: A new algorithm and proof. Syst. Control Lett. 4, 243--251.Google ScholarCross Ref
- Van Loan, C. F. 1982. Using the Hessenberg decomposition in control theory. Math. Program. Study 18, 102--111.Google ScholarCross Ref
Index Terms
- Efficient generalized Hessenberg form and applications
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