Subspace extraction for matrix functions

We investigate existing and novel methods to approximate matrix functions using subspace methods. We analyze a two-sided harmonic Ritz approach and apply this to the extraction from a subspace for matrix functions. We derive all methods in various ways and provide a framework to fit in the techniques.

[1]  M. Hochstenbach,et al.  Two-sided and alternating Jacobi-Davidson , 2001 .

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

[3]  Igor Moret,et al.  Interpolating functions of matrices on zeros of quasi‐kernel polynomials , 2005, Numer. Linear Algebra Appl..

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

[5]  H. V. D. Vorst,et al.  EFFICIENT EXPANSION OF SUBSPACES IN THE JACOBI-DAVIDSON METHOD FOR STANDARD AND GENERALIZED EIGENPROBLEMS , 1998 .

[6]  Henk A. van der Vorst,et al.  Approximate solutions and eigenvalue bounds from Krylov subspaces , 1995, Numer. Linear Algebra Appl..

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

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

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

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

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

[12]  Andreas Stathopoulos,et al.  A Case for a Biorthogonal Jacobi-Davidson Method: Restarting and Correction Equation , 2002, SIAM J. Matrix Anal. Appl..

[13]  D. Sorensen Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[14]  H. V. D. Vorst,et al.  An iterative solution method for solving f ( A ) x = b , using Krylov subspace information obtained for the symmetric positive definite matrix A , 1987 .

[15]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

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

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

[18]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .

[19]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

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