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A multiprecision algorithm for the solution of polynomials and polynomial eigenvalue problems

Published: 28 July 2014 Publication History

Abstract

Many applications of the real world are modelled by matrix polynomials [EQUATION] where Ai are m x m matrices, see for instance [2], [6]. A computational task encountered in this framework is computing the eigenvalues of P(x), that is, the solutions of the polynomial equation det P(x) = 0. This task is generally accomplished by reducing P(x) to a linear pencil of the kind xL -- K for suitable matrices K, L of size mn, and to solving the eigenvalue problem (λL -- K)v = 0 by means of standard numerical algorithms. A wide literature exists on this approach, we refer the reader to [7] for an example.

References

[1]
Dario A. Bini and Giuseppe Fiorentino. Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numerical Algorithms, 23: 127--173, 2000.
[2]
Dario A. Bini, Guy Latouche, and Beatrice Meini. Numerical methods for structured Markov chains. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2005. Oxford Science Publications.
[3]
Dario A. Bini, Vanni Noferini, and Meisam Sharify. Locating the eigenvalues of matrix polynomials. SIAM J. Matrix Anal. Appl., 34(4): 1708--1727, 2013.
[4]
Dario A Bini and Leonardo Robol. Solving secular and polynomial equations: A multiprecision algorithm. Journal of Computational and Applied Mathematics, 2014, http://dx.doi.org/10.1016/j.cam.2013.04.037.
[5]
Fernando De Teran, Froilan M. Dopico, and Steven D. Mackey. Spectral equivalence of matrix polynomials and the index sum theorem. Technical report, MIMS E-Prints, Manchester, 2013.
[6]
D. Steven Mackey, Niloufer Mackey, Christian Mehl, and Volker Mehrmann. Structured polynomial eigenvalue problems: good vibrations from good linearizations. SIAM J. Matrix Anal. Appl., 28(4): 1029--1051 (electronic), 2006.
[7]
D. Steven Mackey, Niloufer Mackey, Christian Mehl, and Volker Mehrmann. Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl., 28(4): 971--1004 (electronic), 2006.
[8]
Raf Vandebril, Marc Van Barel, and Nicola Mastronardi. Matrix computations and semiseparable matrices: linear systems, volume 1. JHU Press, 2007.

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SNC '14: Proceedings of the 2014 Symposium on Symbolic-Numeric Computation
July 2014
154 pages
ISBN:9781450329637
DOI:10.1145/2631948
Permission to make digital or hard copies of part or all 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 third-party components of this work must be honored. For all other uses, contact the Owner/Author.

Sponsors

  • 973 Program: National Basic Research Program of China
  • KLMM: Key Laboratory of Mathematics Mechanization
  • MapleSoft
  • ORCCA: Ontario Research Centre for Computer Algebra
  • NSFC: Natural Science Foundation of China
  • Chinese Academy of Engineering: Chinese Academy of Engineering
  • NAG: Numerical Algorithms Group

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 28 July 2014

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Author Tags

  1. PEP
  2. linearizations
  3. matrix polynomials

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  • Research-article

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SNC '14
Sponsor:
  • 973 Program
  • KLMM
  • ORCCA
  • NSFC
  • Chinese Academy of Engineering
  • NAG
SNC '14: Symbolic-Numeric Computation 2014
July 28 - 31, 2014
Shanghai, China

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