Recovery of Eigenvectors and Minimal Bases of Matrix Polynomials from Generalized Fiedler Linearizations

A standard way to solve polynomial eigenvalue problems P(λ)x=0 is to convert the matrix polynomial P(λ) into a matrix pencil that preserves its elementary divisors and, therefore, its eigenvalues. This process is known as linearization and is not unique, since there are infinitely many linearizations with widely varying properties associated with P(λ). This freedom has motivated the recent development and analysis of new classes of linearizations that generalize the classical first and second Frobenius companion forms, with the goals of finding linearizations that retain whatever structures that P(λ) might possess and/or of improving numerical properties, as conditioning or backward errors, with respect the companion forms. In this context, an important new class of linearizations is what we name generalized Fiedler linearizations, introduced in 2004 by Antoniou and Vologiannidis as an extension of certain linearizations introduced previously by Fiedler for scalar polynomials. On the other hand, the mere ...

[1]  Volker Mehrmann,et al.  Trimmed linearizations for structured matrix polynomials , 2008 .

[2]  R. Byers,et al.  Symplectic, BVD, and Palindromic Approaches to Discrete-Time Control Problems , 2008 .

[3]  I. Gohberg,et al.  General theory of regular matrix polynomials and band Toeplitz operators , 1988 .

[4]  Sabine Zaglmayr,et al.  Eigenvalue problems in SAW-filter simulations. , 2002 .

[5]  Karl Meerbergen,et al.  The Quadratic Eigenvalue Problem , 2001, SIAM Rev..

[6]  Volker Mehrmann,et al.  Vector Spaces of Linearizations for Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[7]  E. Antoniou,et al.  A new family of companion forms of polynomial matrices , 2004 .

[8]  P. Lancaster,et al.  Linearization of matrix polynomials expressed in polynomial bases , 2008 .

[9]  N. Higham,et al.  Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems , 2008 .

[10]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[11]  James Demmel,et al.  The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part II: software and applications , 1993, TOMS.

[12]  Gene H. Golub,et al.  Matrix computations , 1983 .

[13]  James Demmel,et al.  The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part I: theory and algorithms , 1993, TOMS.

[14]  P. Dooren,et al.  The eigenstructure of an arbitrary polynomial matrix : Computational aspects , 1983 .

[15]  Nicholas J. Higham,et al.  Symmetric Linearizations for Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[16]  Wen-Wei Lin,et al.  Structure-Preserving Algorithms for Palindromic Quadratic Eigenvalue Problems Arising from Vibration of Fast Trains , 2008, SIAM J. Matrix Anal. Appl..

[17]  Nicholas J. Higham,et al.  Backward Error of Polynomial Eigenproblems Solved by Linearization , 2007, SIAM J. Matrix Anal. Appl..

[18]  Erik Elmroth,et al.  A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils. Part I: Versal Deformations , 1997, SIAM J. Matrix Anal. Appl..

[19]  Nicholas J. Higham,et al.  The Conditioning of Linearizations of Matrix Polynomials , 2006, SIAM J. Matrix Anal. Appl..

[20]  F. M. Dopico,et al.  Structured Linearizations for Palindromic Matrix Polynomials of Odd Degree , 2010 .

[21]  Volker Mehrmann,et al.  Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations , 2006, SIAM J. Matrix Anal. Appl..

[22]  P. Dooren.  The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[23]  Froilán M. Dopico,et al.  SHARP LOWER BOUNDS FOR THE DIMENSION OF LINEARIZATIONS OF MATRIX POLYNOMIALS , 2008 .

[24]  Jr. G. Forney,et al.  Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems , 1975 .

[25]  U. Langer,et al.  Finite-element simulation of wave propagation in periodic piezoelectric SAW structures , 2006, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[26]  Froilán M. Dopico,et al.  Fiedler Companion Linearizations and the Recovery of Minimal Indices , 2010, SIAM J. Matrix Anal. Appl..

[27]  F. M. Dopico,et al.  LINEARIZATIONS OF SINGULAR MATRIX POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES , 2009 .

[28]  Thomas Kailath,et al.  Linear Systems , 1980 .

[29]  Miroslav Fiedler,et al.  A note on companion matrices , 2003 .

[30]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[31]  Peter Lancaster,et al.  A Note on Weak and Strong Linearizations of Regular Matrix Polynomials , 2005 .

[32]  Volker Mehrmann,et al.  ON THE SOLUTION OF PALINDROMIC EIGENVALUE PROBLEMS , 2004 .

[33]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.