On Some Modiications of the Lanczos Algorithm and the Relation with Pad E Approximations

In this paper we try to show the relations between the Lanczos algorithm and Pad e approximations as used e.g. in identiication and model reduction of dynamical systems. We also explore the use of variants of the Lanczos method in order to obtain approximations with better properties than the ones resulting from the standard Lanczos algorithm.

[1]  Imad M. Jaimoukha,et al.  Oblique Production Methods for Large Scale Model Reduction , 1995, SIAM J. Matrix Anal. Appl..

[2]  L. Spaanenburg,et al.  PROCEEDINGS OF THE IEEE 1995 CUSTOM INTEGRATED CIRCUITS CONFERENCE , 1995 .

[3]  Daniel Boley.  Krylov space methods on state-space control models , 1994 .

[4]  P. Dooren,et al.  Asymptotic Waveform Evaluation via a Lanczos Method , 1994 .

[5]  E. Grimme,et al.  Stable partial realizations via an implicitly restarted Lanczos method , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[6]  Axel Ruhe.  The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices , 1994 .

[7]  Axel Ruhe.  Rational Krylov Algorithms for Nonsymmetric Eigenvalue Problems , 1994 .

[8]  Axel Ruhe.  Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. matrix pairs , 1994 .

[9]  Roland W. Freund,et al.  An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices , 1993, SIAM J. Sci. Comput..

[10]  M Maarten Steinbuch,et al.  Robust control of a compact disc player , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[11]  Yousef Saad,et al.  Efficient Solution of Parabolic Equations by Krylov Approximation Methods , 1992, SIAM J. Sci. Comput..

[12]  Beresford N. Parlett,et al.  Reduction to Tridiagonal Form and Minimal Realizations , 1992, SIAM J. Matrix Anal. Appl..

[13]  A. Laub,et al.  Frequency response computation via rational interpolation , 1992, IEEE Symposium on Computer-Aided Control System Design.

[14]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

[15]  Martin H. Gutknecht,et al.  A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms, Part I , 1992, SIAM J. Matrix Anal. Appl..

[16]  Roy R. Craig,et al.  An unsymmetric Lanczos algorithm for damped structural dynamics systems , 1992 .

[17]  BO K Agstr,et al.  A GENERALIZED STATE-SPACE APPROACH FOR THE ADDITIVE DECOMPOSITION OF A TRANSFER MATRIX , 1992 .

[18]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[19]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[20]  Roy R. Craig,et al.  Computational enhancement of an unsymmetric block Lanczos algorithm , 1990 .

[21]  Brian D. O. Anderson,et al.  Rational interpolation and state-variable realizations , 1990 .

[22]  Lawrence T. Pileggi,et al.  Asymptotic waveform evaluation for timing analysis , 1990, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[23]  Roy R. Craig,et al.  Structural dynamics analysis using an unsymmetric block Lanczos algorithm , 1988 .

[24]  Robert Skelton,et al.  Model reductions using a projection formulation , 1987, 26th IEEE Conference on Decision and Control.

[25]  Xiheng Hu,et al.  FF-Padé method of model reduction in frequency domain , 1987 .

[26]  Zhishun A. Liu,et al.  A Look Ahead Lanczos Algorithm for Unsymmetric Matrices , 1985 .

[27]  W. Gragg,et al.  On the partial realization problem , 1983 .

[28]  Anders Lindquist,et al.  The stability and instability of partial realizations , 1982 .

[29]  M. A. Brebner,et al.  Eigenvalues of Ax = λBx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process , 1982 .

[30]  A. Bunse-Gerstner.  An analysis of the HR algorithm for computing the eigenvalues of a matrix , 1981 .

[31]  C. Brezinski.  Padé-type approximation and general orthogonal polynomials , 1980 .

[32]  Y. Shamash.  Model reduction using the Routh stability criterion and the Padé approximation technique , 1975 .

[33]  W. Gragg,et al.  The Padé Table and Its Relation to Certain Algorithms of Numerical Analysis , 1972 .

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

[35]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .