A new investigation of the extended Krylov subspace method for matrix function evaluations

For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  Marlis Hochbruck,et al.  Exponential Integrators for Large Systems of Differential Equations , 1998, SIAM J. Sci. Comput..

[2]  L. Trefethen,et al.  Talbot quadratures and rational approximations , 2006 .

[3]  Anne Greenbaum,et al.  The polynomial numerical hulls of Jordan blocks and related matrices , 2003 .

[4]  J. Boyle,et al.  HSL MI 20 : an efficient AMG preconditioner , 2007 .

[5]  L. Knizhnerman,et al.  Two polynomial methods of calculating functions of symmetric matrices , 1991 .

[6]  P. Petrushev,et al.  Rational Approximation of Real Functions , 1988 .

[7]  Kamy Sepehrnoori,et al.  An extended Krylov subspace method to simulate single-phase fluid flow phenomena in axisymmetric and anisotropic porous media , 2003 .

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

[9]  Tobin A. Driscoll,et al.  Algorithm 756: a MATLAB toolbox for Schwarz-Christoffel mapping , 1996, TOMS.

[10]  Anthony W. Thomas,et al.  Nucleon Mass In Chiral Effective Field Theory , 2005 .

[11]  Marlis Hochbruck,et al.  Exponential Integrators for Quantum-Classical Molecular Dynamics , 1999 .

[12]  P. Sonneveld,et al.  IDR(s): A family of simple and fast algorithms for solving large nonsymmetric linear systems , 2007 .

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

[14]  Igor Moret.  Rational Lanczos approximations to the matrix square root and related functions , 2009, Numer. Linear Algebra Appl..

[15]  Valeria Simoncini,et al.  Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing , 2003, SIAM J. Sci. Comput..

[16]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[17]  M. Hochbruck,et al.  Exponential Runge--Kutta methods for parabolic problems , 2005 .

[18]  H. V. D. Vorst,et al.  Numerical methods for the QCDd overlap operator. I. Sign-function and error bounds , 2002, hep-lat/0202025.

[19]  H. Tal-Ezer.  Spectral methods in time for parabolic problems , 1989 .

[20]  Michel Crouzeix,et al.  Numerical range and functional calculus in Hilbert space , 2007 .

[21]  Andreas Frommer,et al.  QCD and numerical analysis III : proceedings of the Third International Workshop on Numerical Analysis and Lattice QCD, Edinburgh, June-July 2003 , 2005 .

[22]  Andreas Frommer,et al.  MONOTONE CONVERGENCE OF THE LANCZOS APPROXIMATIONS TO MATRIX FUNCTIONS OF HERMITIAN MATRICES , 2009 .

[23]  Christof Schütte,et al.  Numerical Integrators for Quantum-Classical Molecular Dynamics , 1999, Computational Molecular Dynamics.

[24]  Vladimir Druskin,et al.  Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic , 1995, Numer. Linear Algebra Appl..

[25]  P. K. Suetin.  Series of Faber polynomials , 1998 .

[26]  Oliver G. Ernst,et al.  A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions , 2006, SIAM J. Numer. Anal..

[27]  A. Boriçi,et al.  Computational methods for UV-suppressed fermions , 2003 .

[28]  I. Moret,et al.  An interpolatory approximation of the matrix exponential based on Faber polynomials , 2001 .

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

[30]  ANDREAS FROMMER,et al.  Stopping Criteria for Rational Matrix Functions of Hermitian and Symmetric Matrices , 2008, SIAM J. Sci. Comput..

[31]  L. Tuckerman,et al.  A method for exponential propagation of large systems of stiff nonlinear differential equations , 1989 .

[32]  Martin B. van Gijzen,et al.  IDR(s): A Family of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear Equations , 2008, SIAM J. Sci. Comput..

[33]  Vladimir Druskin,et al.  On monotonicity of the Lanczos approximation to the matrix exponential , 2008 .

[34]  Valeria Simoncini,et al.  Analysis of Projection Methods for Rational Function Approximation to the Matrix Exponential , 2006, SIAM J. Numer. Anal..

[35]  Anne Greenbaum,et al.  Using Nonorthogonal Lanczos Vectors in the Computation of Matrix Functions , 1998, SIAM J. Sci. Comput..

[36]  Vladimir Druskin,et al.  New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry , 1999 .

[37]  K. Gustafson,et al.  Numerical Range: The Field Of Values Of Linear Operators And Matrices , 1996 .

[38]  Valeria Simoncini,et al.  A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations , 2007, SIAM J. Sci. Comput..

[39]  L. Knizhnerman.  Calculation of functions of unsymmetric matrices using Arnoldi's method , 1991 .

[40]  Luciano Lopez,et al.  Acceleration techniques for approximating the matrix exponential , 2008 .

[41]  Sergey I. Vinitsky,et al.  Magnus-factorized method for numerical solving the time-dependent Schrödinger equation , 2000 .

[42]  Yousef Saad,et al.  A Parallel Block Cyclic Reduction Algorithm for the Fast Solution of Elliptic Equations , 1987, ICS.

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

[44]  Valeria Simoncini,et al.  Acceleration Techniques for Approximating the Matrix Exponential Operator , 2008, SIAM J. Matrix Anal. Appl..

[45]  Bernhard Beckermann,et al.  Image numérique, GMRES et polynômes de Faber , 2005 .

[46]  Andr'e Nauts,et al.  New Approach to Many-State Quantum Dynamics: The Recursive-Residue-Generation Method , 1983 .

[47]  Marco Vianello,et al.  Efficient computation of the exponential operator for large, sparse, symmetric matrices , 2000, Numer. Linear Algebra Appl..

[48]  Igor Moret,et al.  Error Estimates for Polynomial Krylov Approximations to Matrix Functions , 2008, SIAM J. Matrix Anal. Appl..

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

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

[51]  L. Knizhnerman,et al.  Spectral approach to solving three-dimensional Maxwell's diffusion equations in the time and frequency domains , 1994 .

[52]  I. Turner,et al.  A restarted Lanczos approximation to functions of a symmetric matrix , 2010 .