Implicit QR algorithms for palindromic and even eigenvalue problems

In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques.

[1]  K. Dasgupta,et al.  Matrix perturbation theory for M-theory on a PP-wave , 2002, hep-th/0205185.

[2]  Leiba Rodman,et al.  Bounded and stably bounded palindromic difference equations of first order , 2006 .

[3]  David S. Watkins,et al.  Theory of Decomposition and Bulge-Chasing Algorithms for the Generalized Eigenvalue Problem , 1994 .

[4]  Roger A. Horn,et al.  A regularization algorithm for matrices of bilinear and sesquilinear forms , 2006 .

[5]  Bo Kågström,et al.  Computing eigenspaces with specified eigenvalues of a regular matrix pair (A, B) and condition estimation: theory, algorithms and software , 1996, Numerical Algorithms.

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

[7]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[8]  Werner Rheinboldt,et al.  Computer Science and Scientific Computing , 1989 .

[9]  Roger A. Horn,et al.  Canonical forms for complex matrix congruence and ∗congruence , 2006, 0709.2473.

[10]  Ludwig Elsner,et al.  Perturbation thèorems for the generalized eigenvalue problem , 1982 .

[11]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[12]  Daniel B. Szyld,et al.  The matrix eigenvalue problem: GR and Krylov subspace methods , 2009, Math. Comput..

[13]  R. Byers.  A Hamiltonian $QR$ Algorithm , 1986 .

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

[15]  Karen S. Braman,et al.  The Multishift QR Algorithm. Part I: Maintaining Well-Focused Shifts and Level 3 Performance , 2001, SIAM J. Matrix Anal. Appl..

[16]  G. Stewart.  On the Sensitivity of the Eigenvalue Problem $Ax = \lambda Bx$ , 1972 .

[17]  Daniel Kressner,et al.  Numerical Methods for General and Structured Eigenvalue Problems , 2005, Lecture Notes in Computational Science and Engineering.

[18]  J. Demmel,et al.  On swapping diagonal blocks in real Schur form , 1993 .

[19]  David S. Watkins,et al.  The transmission of shifts and shift blurring in the QR algorithm , 1996 .

[20]  E. Chu,et al.  Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms , 2008 .

[21]  Daniel Kressner,et al.  Structured Hölder Condition Numbers for Multiple Eigenvalues , 2009, SIAM J. Matrix Anal. Appl..

[22]  C. Schröder.  URV decomposition based structured methods for palindromic and even eigenvalue problems , 2007 .

[23]  Volker Mehrmann,et al.  A symplectic orthogonal method for single input or single ouput discrete time optimal quadratic control problems , 1988 .

[24]  Karen S. Braman,et al.  The Multishift QR Algorithm. Part II: Aggressive Early Deflation , 2001, SIAM J. Matrix Anal. Appl..

[25]  Daniel Kressner,et al.  Structured Condition Numbers for Invariant Subspaces , 2006, SIAM J. Matrix Anal. Appl..

[26]  G. W. Stewart,et al.  Computer Science and Scientific Computing , 1990 .

[27]  Daniel Kressner,et al.  Multishift Variants of the QZ Algorithm with Aggressive Early Deflation , 2006, SIAM J. Matrix Anal. Appl..

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

[29]  David S. Watkins,et al.  Convergence of algorithms of decomposition type for the eigenvalue problem , 1991 .

[30]  R. C. Thompson,et al.  Pencils of complex and real symmetric and skew matrices , 1991 .

[31]  S. Rump.  EIGENVALUES, PSEUDOSPECTRUM AND STRUCTURED PERTURBATIONS , 2006 .

[32]  Volker Mehrmann,et al.  Numerical methods for palindromic eigenvalue problems: Computing the anti‐triangular Schur form , 2009, Numer. Linear Algebra Appl..

[33]  David S. Watkins.  QR-like Algorithms--- An Overview of Convergence Theory and Practice , 1996 .

[34]  Christian Mehl,et al.  On Asymptotic Convergence of Nonsymmetric Jacobi Algorithms , 2008, SIAM J. Matrix Anal. Appl..

[35]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[36]  R. C. Thompson,et al.  The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil , 1976 .

[37]  Doktor der Naturwissenschaften.  Palindromic and Even Eigenvalue Problems - Analysis and Numerical Methods , 2008 .