Generalized Rational Krylov Decompositions with an Application to Rational Approximation

Generalized rational Krylov decompositions are matrix relations which, under certain conditions, are associated with rational Krylov spaces. We study the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation, and two algorithms are presented for this task. Using such transformations we develop a rational Krylov method for rational least squares fitting. Numerical experiments indicate that the proposed method converges fast and robustly. A MATLAB toolbox with implementations of the presented algorithms and experiments is provided.

[1]  Lothar Reichel,et al.  Recursion relations for the extended Krylov subspace method , 2011 .

[2]  Raf Vandebril,et al.  Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices , 2014 .

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

[4]  Wim Michiels,et al.  NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems , 2014, SIAM J. Sci. Comput..

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

[6]  Marlis Hochbruck,et al.  Preconditioning Lanczos Approximations to the Matrix Exponential , 2005, SIAM J. Sci. Comput..

[7]  L. Knizhnerman,et al.  Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions , 1998, SIAM J. Matrix Anal. Appl..

[8]  Zlatko Drmac,et al.  Quadrature-Based Vector Fitting: Implications For H2 System Approximation , 2014, ArXiv.

[9]  Axel Ruhe.  The Rational Krylov Algorithm for Nonlinear Matrix Eigenvalue Problems , 2003 .

[10]  A. Antoulas,et al.  A Rational Krylov Iteration for Optimal H 2 Model Reduction , 2006 .

[11]  T. Dhaene,et al.  A comparative study of vector fitting and orthonormal vector fitting techniques for EMC applications , 2006, 2006 IEEE International Symposium on Electromagnetic Compatibility, 2006. EMC 2006..

[12]  Adhemar Bultheel,et al.  Orthogonal Rational Functions , 1999, Cambridge monographs on applied and computational mathematics.

[13]  S. Güttel.  Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection , 2013 .

[14]  Axel Ruhe,et al.  Rational Krylov: A Practical Algorithm for Large Sparse Nonsymmetric Matrix Pencils , 1998, SIAM J. Sci. Comput..

[15]  Volker Grimm,et al.  Uniform Approximation of φ-Functions in Exponential Integrators by a Rational Krylov Subspace Method with Simple Poles , 2014, SIAM J. Matrix Anal. Appl..

[16]  Lothar Reichel,et al.  Error Estimates and Evaluation of Matrix Functions via the Faber Transform , 2009, SIAM J. Numer. Anal..

[17]  Karl Meerbergen,et al.  Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem , 2001, Numer. Linear Algebra Appl..

[18]  B. Gustavsen,et al.  Improving the pole relocating properties of vector fitting , 2006, 2006 IEEE Power Engineering Society General Meeting.

[19]  Adhemar Bultheel,et al.  Using implicitly filtered RKS for generalised eigenvalue problems , 1999 .

[20]  G. W. Stewart,et al.  Matrix algorithms , 1998 .

[21]  Raf Vandebril,et al.  A Generalization of the Multishift QR Algorithm , 2012, SIAM J. Matrix Anal. Appl..

[22]  H. Voss,et al.  Rational Krylov for Nonlinear Eigenproblems, an Iterative Projection Method , 2005 .

[23]  Lothar Reichel,et al.  Rational Gauss Quadrature , 2014, SIAM J. Numer. Anal..

[24]  G. W. Stewart,et al.  Addendum to "A Krylov-Schur Algorithm for Large Eigenproblems" , 2002, SIAM J. Matrix Anal. Appl..

[25]  Adhemar Bultheel,et al.  RATIONAL KRYLOV SEQUENCES AND ORTHOGONAL RATIONAL FUNCTIONS , 2008 .

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

[27]  A. Semlyen,et al.  Rational approximation of frequency domain responses by vector fitting , 1999 .

[28]  Stefan Güttel,et al.  Three-Dimensional Transient Electromagnetic Modeling Using Rational Krylov Methods , 2015 .

[29]  Daniel Kressner,et al.  Block algorithms for reordering standard and generalized Schur forms , 2006, TOMS.

[30]  Lloyd N. Trefethen,et al.  Robust Padé Approximation via SVD , 2013, SIAM Rev..

[31]  Dario Fasino.  Rational Krylov matrices and QR steps on Hermitian diagonal-plus-semiseparable matrices , 2005, Numer. Linear Algebra Appl..

[32]  Daniel Skoogh,et al.  Rational Krylov Algorithms for Eigenvalue Computation and Model Reduction , 1998, PARA.

[33]  I. Moret,et al.  RD-Rational Approximations of the Matrix Exponential , 2004 .

[34]  Paul Van Dooren,et al.  A rational Lanczos algorithm for model reduction , 1996, Numerical Algorithms.

[35]  Dario Fasino,et al.  Orthogonal Rational Functions and Structured Matrices , 2005, SIAM J. Matrix Anal. Appl..

[36]  Stefan Güttel,et al.  The RKFIT Algorithm for Nonlinear Rational Approximation , 2017, SIAM J. Sci. Comput..

[37]  B. Gustavsen,et al.  Comments on “a comparative study of vector fitting and orthonormal vector fitting techniques for EMC applications” , 2007, 2007 18th International Zurich Symposium on Electromagnetic Compatibility.

[38]  Vladimir Druskin,et al.  Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts , 2009, SIAM J. Sci. Comput..

[39]  G. W. Stewart,et al.  A Krylov-Schur Algorithm for Large Eigenproblems , 2001, SIAM J. Matrix Anal. Appl..

[40]  Zvonimir Bujanovic,et al.  A new framework for implicit restarting of the Krylov–Schur algorithm , 2015, Numer. Linear Algebra Appl..

[41]  T. Dhaene,et al.  Orthonormal Vector Fitting: A Robust Macromodeling Tool for Rational Approximation of Frequency Domain Responses , 2007, IEEE Transactions on Advanced Packaging.

[42]  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.

[43]  Stefan Güttel,et al.  Rational Krylov Methods for Operator Functions , 2010 .

[44]  L. Trefethen,et al.  Robust rational interpolation and least-squares , 2011 .

[45]  Adhemar Bultheel,et al.  Orthogonal basis functions in discrete least-squares rational approximation , 2004 .

[46]  Stefan Güttel,et al.  A Rational Krylov Toolbox for MATLAB , 2014 .

[47]  Axel Ruhe.  Rational Krylov sequence methods for eigenvalue computation , 1984 .

[48]  Wim Michiels,et al.  A Rational Krylov Method Based on Hermite Interpolation for Nonlinear Eigenvalue Problems , 2012, SIAM J. Sci. Comput..

[49]  K. Meerbergen,et al.  The implicit application of a rational filter in the RKS method , 1997 .