Approximating the leading singular triplets of a large matrix function

Abstract Given a large square matrix A and a sufficiently regular function f so that f ( A ) is well defined, we are interested in the approximation of the leading singular values and corresponding left and right singular vectors of f ( A ) , and in particular in the approximation of ‖ f ( A ) ‖ , where ‖ ⋅ ‖ is the matrix norm induced by the Euclidean vector norm. Since neither f ( A ) nor f ( A ) v can be computed exactly, we introduce a new inexact Golub–Kahan–Lanczos bidiagonalization procedure, where the inexactness is related to the inaccuracy of the operations f ( A ) v , f ( A ) ⁎ v . Particular outer and inner stopping criteria are devised so as to cope with the lack of a true residual. Numerical experiments with the new algorithm on typical application problems are reported.

[1]  Charles William Gear,et al.  Numerical Solution of Ordinary Differential Equations: Is There Anything Left to Do? , 1981 .

[2]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

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

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

[5]  Michele Benzi,et al.  Decay Properties of Spectral Projectors with Applications to Electronic Structure , 2012, SIAM Rev..

[6]  Z. Strakos,et al.  Krylov Subspace Methods: Principles and Analysis , 2012 .

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

[8]  Emanuel H. Rubensson.  Controlling Errors in Recursive Fermi-Dirac Operator Expansions with Applications in Electronic Structure Theory , 2012, SIAM J. Sci. Comput..

[9]  R. Larsen.  Lanczos Bidiagonalization With Partial Reorthogonalization , 1998 .

[10]  Valérie Frayssé,et al.  Inexact Matrix-Vector Products in Krylov Methods for Solving Linear Systems: A Relaxation Strategy , 2005, SIAM J. Matrix Anal. Appl..

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

[12]  Stefan Güttel,et al.  Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices , 2014, SIAM J. Matrix Anal. Appl..

[13]  B. Baxter.  Norm Estimates for Inverses of Toeplitz Distance Matrices , 1994 .

[14]  L. Trefethen,et al.  Spectra and pseudospectra : the behavior of nonnormal matrices and operators , 2005 .

[15]  G. Watson.  Computing the numerical radius , 1996 .

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

[17]  D. Choi.  Estimating Norms of Matrix Functions using Numerical Ranges , 2013 .

[18]  Nicholas J. Higham,et al.  Estimating the Condition Number of the Fréchet Derivative of a Matrix Function , 2014, SIAM J. Sci. Comput..

[19]  L. Trefethen.  Spectra and pseudospectra , 2005 .

[20]  M. Overton,et al.  Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix , 2005 .

[21]  Gene H. Golub,et al.  Calculating the singular values and pseudo-inverse of a matrix , 2007, Milestones in Matrix Computation.

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

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

[24]  Valeria Simoncini,et al.  Variable Accuracy of Matrix-Vector Products in Projection Methods for Eigencomputation , 2005, SIAM J. Numer. Anal..

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

[26]  J. Cullum,et al.  Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

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

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

[29]  Emanuel H. Rubensson,et al.  On the condition number and perturbation of matrix functions for Hermitian matrices , 2012, 1206.1762.

[30]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[31]  Valeria Simoncini,et al.  A new investigation of the extended Krylov subspace method for matrix function evaluations , 2009, Numer. Linear Algebra Appl..

[32]  Michael I. Gil.  PERTURBATIONS OF FUNCTIONS OF DIAGONALIZABLE MATRICES , 2010 .

[33]  R. Bhatia,et al.  Norm inequalities related to the matrix geometric mean , 2012, 1502.04497.

[34]  B. A. Schmitt.  Norm bounds for rational matrix functions , 1983 .

[35]  J. Cullum,et al.  Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1 , 2002 .

[36]  Z. Strakos,et al.  On error estimation in the conjugate gradient method and why it works in finite precision computations. , 2002 .

[37]  M. Hochbruck,et al.  Exponential integrators , 2010, Acta Numerica.

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

[39]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

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

[41]  N. Higham.  Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics) , 2008 .