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Structured Linearizations for Matrix Polynomials

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Structured Linearizations for Matrix Polynomials Structured Linearizations for Matrix Polynomials Mackey, D. Steven 2006 MIMS EPrint: 2006.68 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 STRUCTURED LINEARIZATIONS FOR MATRIX POLYNOMIALS A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2006 D. Steven Mackey School of Mathematics Contents Abstract 6 Declaration 7 Copyright 8 Publications 9 Advisor and Examiners 10 Acknowledgements 11 1 Introduction 13 1.1 Some Preliminaries . 16 2 A Vector Space Setting for Linearizations 19 2.1 Vector Spaces of \Potential" Linearizations . 19 2.2 When is a Pencil in L1(P ) a Linearization? . 25 2.2.1 The Strong Linearization Theorem . 25 2.2.2 Linearization Conditions . 29 2.3 Another View of L1(P ) and L2(P ) ................... 31 3 DL(P ) and Block-symmetry 33 3.1 The Block-transpose Operation . 33 3.2 Block-symmetry and Shifted Sum Equations . 36 3.2.1 Shifted Sum Equations . 37 3.3 Block-symmetric Pencils in L1(P ).................... 39 3.3.1 The Subspace B(P )........................ 40 3.3.2 The \Standard Basis" for B(P ) . 40 3.4 Double Ansatz Pencils for General P . 44 3.5 Other Block-symmetric Linearizations . 45 4 The Genericity of Linearizations in DL(P ) 48 4.1 Some Suggestive Examples . 48 4.2 Determinant of DL(P )-pencils . 50 4.3 The Eigenvalue Exclusion Theorem . 59 2 5 Symmetric and Hermitian Linearizations 62 5.1 Symmetric Pencils in L1(P ) for Symmetric P . 62 5.2 Hermitian Pencils in L1(P ) for Hermitian P . 63 5.3 Genericity of Linearizations in S(P ) and H(P ) . 64 6 Palindromic and Alternating Polynomials 66 6.1 Basic Structures: De¯nitions and Notation . 66 6.2 Spectral Symmetry . 69 6.3 Cayley Transformations . 72 6.4 Applications . 74 6.5 A Palindrome Sampler . 80 7 Structured Linearizations 82 7.1 A Simple Example . 83 7.2 Existence of Structured Pencils in L1(P ) . 84 7.3 Construction of Structured Pencils . 87 7.3.1 Construction Using the Standard Basis for DL(P ) . 87 7.3.2 Structured Pencils via Shifted Sums . 88 7.4 Which Structured Pencils are Linearizations? . 90 7.5 When Pairings Degenerate . 92 7.6 Good Vibrations from Good Linearizations . 95 7.7 Structured Subspaces of L1(P )...................... 95 8 Conditioning of Eigenvalues of Linearizations 99 8.1 Eigenvalue Conditioning of Matrix Polynomials . 99 8.2 Eigenvalue Conditioning of DL(P )-Linearizations . 102 8.3 Minimizing ·bL(®; ¯) and ·L(¸) . 104 8.3.1 Several Eigenvalues . 108 8.4 Quadratic Polynomials . 109 8.5 Linearizing the Reversal of P . 110 8.6 Companion Linearizations . 112 8.7 Numerical Experiments . 115 8.8 Addendum | Some Useful Bounds . 121 Bibliography 123 3 List of Tables 3.3.1 \Standard basis pencils" in B(Q) for quadratic Q(¸) = ¸2A + ¸B + C. 43 3.3.2 \Standard basis pencils" in B(P ) for cubic P (¸) = ¸3A + ¸2B + ¸C + D. 43 4.1.1 Some pencils in DL(P ) for the general quadratic P (¸) = ¸2A+¸B+C. Linearization condition found using procedure in Section 2.2.2. 49 4.1.2 Some pencils in DL(P ) for the general cubic P (¸) = ¸3A+¸2B +¸C + D. Linearization condition found using procedure in Section 2.2.2. 49 6.1.1 De¯nitions of basic structures . 67 6.2.1 Spectral symmetries . 69 6.3.1 Cayley transformations of structured matrix polynomials . 73 7.2.1 Admissible ansatz vectors for structured pencils (palindromic structures) 85 7.2.2 Admissible ansatz vectors for structured pencils (alternating structures) 87 7.4.1 Parallelism of Structures . 92 7.7.1 Structured linearizations for P (¸) = ¸2A + ¸B + C. Except for the parameters r 2 R and z 2 C, the linearizations are unique up to a (suitable) scalar factor. The last column lists the roots of the v- polynomial p(x ; w) corresponding to w = Rv (for palindromic struc- tures) or w = §v (for alternating structures); for L to be a linearization these eigenvalues must be excluded from P . 97 7.7.2 ? -palindromic linearizations for the ?-palindromic matrix polynomial P (¸) = ¸3A + ¸2B + ¸B? + A?. The last column lists the roots of the v-polynomial corresponding to Rv. All ? -palindromic linearizations in L1(P ) for this matrix polynomial are linear combinations of the ¯rst two linearizations in the case ? = T , and real linear combinations of the ¯rst three linearizations in the case ? = ¤. A speci¯c example is given by the fourth linearization. 98 7.7.3 ? -even linearizations for the ? -even matrix polynomial P (¸) = ¸3A + ¸2B + ¸C + D, where A = ¡A?, B = B?, C = ¡C?, D = D?. The last column lists the roots of the v-polynomial corresponding to §v. All ? -even linearizations in L1(P ) for this matrix polynomial are linear combinations of the ¯rst two linearizations in the case ? = T , and real linear combinations of the ¯rst three linearizations in the case ? = ¤. A speci¯c example is given by the fourth linearization. 98 8.7.1 Problem statistics. 116 4 List of Figures 6.4.1 FE discretization of the rail in one sleeper bay. 74 8.7.1 Wave problem, unscaled; v = e1, ½ = 625. 118 8.7.2 Wave problem, scaled; v = e2, ½ = 1. 118 4 8.7.3 Nuclear power plant problem, unscaled; v = e1, ½ = 7 £ 10 . 119 8.7.4 Nuclear power plant problem, scaled; v = e2, ½ = 1. 119 8.7.5 Damped mass-spring system, unscaled; v = e1, ½ = 320. 120 8.7.6 Damped mass-spring system, unscaled; v = e2, ½ = 320. 120 5 Abstract The classical approach to investigating polynomial eigenvalue problems is lineariza- tion, where the underlying matrix polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are in¯nitely many lineariza- tions with widely varying properties, but in practice the companion forms are typi- cally used. However, these companion forms are not always entirely satisfactory, and linearizations with special properties may sometimes be required. Given a matrix polynomial P , we develop a systematic approach to generating large classes of linearizations for P . We show how to simply construct two vector spaces of pencils that generalize the companion forms of P , and prove that almost all of these pencils are linearizations for P . Eigenvectors of these pencils are shown to be closely related to those of P . A distinguished subspace, denoted DL(P ), is then isolated, and the special properties of these pencils are investigated. These spaces of pencils provide a convenient arena in which to look for structured linearizations of structured polynomials, as well as to try to optimize the conditioning of linearizations. Many applications give rise to nonlinear eigenvalue problems with an underly- ing structured matrix polynomial; perhaps the most well-known are symmetric and Hermitian polynomials. In this thesis we also identify several less well-known types of structured polynomial (e.g., palindromic, even, odd), explore the relationships between them, and illustrate their appearance in a variety of applications. Special classes of linearizations that reflect the structure of these polynomials, and therefore preserve symmetries in their spectra, are introduced and investigated. We analyze the existence and uniqueness of such linearizations, and show how they may be sys- tematically constructed. The in¯nitely many linearizations of any given polynomial P can have widely varying eigenvalue condition numbers. We investigate the conditioning of lineariza- tions from DL(P ), looking for the best conditioned linearization in that space and comparing its conditioning with that of the original polynomial. We also analyze the eigenvalue conditioning of the widely used ¯rst and second companion linearizations, and ¯nd that they can potentially be much more ill conditioned than P . Our results are phrased in terms of both the standard relative condition number and the condi- tion number of Dedieu and Tisseur for the problem in homogeneous form, this latter condition number having the advantage of applying to zero and in¯nite eigenvalues. 6 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or quali¯cation of this or any other university or other institution of learning. 7 Copyright Copyright in text of this thesis rests with the Author. Copies (by any process) either in full, or of extracts, may be made only in accordance with instructions given by the Author and lodged in the John Rylands University Library of Manchester. Details may be obtained from the Librarian. This page must form part of any such copies made. Further copies (by any process) of copies made in accordance with such instructions may not be made without the permission (in writing) of the Author. The ownership of any intellectual property rights which may be described in this thesis is vested in the University of Manchester, subject to any prior agreement to the contrary, and may not be made available for use by third parties without the written permission of the University, which will prescribe the terms and conditions of any such agreement. Further information on the conditions under which disclosures and exploitation may take place is available from the Head of the School of Mathematics. 8 Publications The work in this thesis is based on the contents of four publications. Chapters 2 and 4 are based on the paper \Vector Spaces of Linearizations for Matrix Polynomials" [58] (with N. Mackey, C. Mehl, and V. Mehrmann), to appear in SIAM Journal of Matrix Analysis and Applications.
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