A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC / SYMMETRIC

We present structure preserving algorithms for the numerical computation of structured staircase forms of skew-symmetric/symmetric matrix pencils along with the Kronecker indices of the associated skew-symmetric/symmetric Kronecker-like canonical form. These methods allow deflation of the singular structure and deflation of infinite eigenvalues with index greater than one. Two algorithms are proposed: one for general skew-symmetric/symmetric pencils and one for pencils in which the skew-symmetric matrix is a direct sum of and . We show how to use the structured staircase form to solve boundary value problems arising in control applications and present numerical examples.

[1]  Christian H. Bischof,et al.  Algorithm 782: codes for rank-revealing QR factorizations of dense matrices , 1998, TOMS.

[2]  V. Mehrmann,et al.  Canonical forms for linear differential-algebraic equations with variable coefficients , 1994 .

[3]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[4]  P. Lancaster,et al.  The Algebraic Riccati Equation , 1995 .

[5]  Peter Benner,et al.  A robust numerical method for the γ-iteration in H∞ control , 2007 .

[6]  Alan J. Laub,et al.  Numerically Reliable Computation of Optimal Performance in Singular $H_{\inf}$ Control , 1997 .

[7]  Doktor der Naturwissenschaften.  Numerical Methods and Software for General and Structured Eigenvalue Problems , 2004 .

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

[9]  V. Mehrmann,et al.  Palindromic Polynomial Eigenvalue Problems:Good Vibrations from Good Linearizations , 2006 .

[10]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[11]  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..

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

[13]  Peter Benner,et al.  Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils , 2002, SIAM J. Matrix Anal. Appl..

[14]  R. Byers,et al.  Descriptor Systems Without Controllability at Infinity , 1997 .

[15]  David S. Watkins,et al.  POLYNOMIAL EIGENVALUE PROBLEMS WITH HAMILTONIAN STRUCTURE , 2002 .

[16]  P. Dooren.  A Generalized Eigenvalue Approach for Solving Riccati Equations , 1980 .

[17]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

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

[19]  Volker Mehrmann,et al.  Numerical methods in control , 2000 .

[20]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[21]  Christian Mehl,et al.  Condensed Forms for Skew-Hamiltonian/Hamiltonian Pencils , 1999, SIAM J. Matrix Anal. Appl..

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

[23]  Mihail M. Konstantinov,et al.  Computational methods for linear control systems , 1991 .

[24]  Volker Mehrmann,et al.  A numerical method for computing the Hamiltonian Schur form , 2006, Numerische Mathematik.

[25]  V. Mehrmann,et al.  Structured Jordan canonical forms for structured matrices that are hermitian, skew hermitian or unitary with respect to indefinite inner products , 1999 .

[26]  Vasile Sima,et al.  Algorithms for Linear-Quadratic Optimization , 2021 .

[27]  V. Mehrmann,et al.  A new method for computing the stable invariant subspace of a real Hamiltonian matrix , 1997 .

[28]  Volker Mehrmann,et al.  A new look at pencils of matrix valued functions , 1994 .

[29]  Peter Kunkel,et al.  Symmetric collocation methods for linear differential-algebraic boundary value problems , 2002, Numerische Mathematik.

[30]  A. Laub,et al.  Numerical solution of the discrete-time periodic Riccati equation , 1994, IEEE Trans. Autom. Control..

[31]  Adam W. Bojanczyk,et al.  Periodic Schur decomposition: algorithms and applications , 1992, Optics & Photonics.

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

[33]  G. Stewart.  The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators , 1980 .

[34]  Volker Mehrmann,et al.  Canonical forms for Hamiltonian and symplectic matrices and pencils , 1999 .

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