Towards backward perturbation bounds for approximate dual Krylov subspaces

Given a matrix A,n by n, and two subspaces K and L of dimension m, we consider how to determine a backward perturbation E whose norm is as small as possible, such that k and L are Krylov subspaces of A+E and its adjoint, respectively. We first focus on determining a perturbation matrix for a given pair of biorthonormal bases, and then take into account how to choose an appropriate biorthonormal pair and express the Krylov residuals as a perturbation of the matrix A. Specifically, the perturbation matrix is globally optimal when A is Hermitian and K=L. The results show that the norm of the perturbation matrix can be assessed by using the norms of the Krylov residuals and those of the biorthonormal bases. Numerical experiments illustrate the efficiency of our strategy.

[1]  Zdenek Strakos,et al.  On Efficient Numerical Approximation of the Bilinear Form c*A-1b , 2011, SIAM J. Sci. Comput..

[2]  W. Kahan,et al.  Residual Bounds on Approximate Eigensystems of Nonnormal Matrices , 1982 .

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

[4]  Jack Dongarra,et al.  Templates for the Solution of Algebraic Eigenvalue Problems , 2000, Software, environments, tools.

[5]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[6]  Javad Mohammadpour,et al.  Efficient modeling and control of large-scale systems , 2010 .

[7]  C. Paige.  Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem , 1980 .

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

[9]  Jack Dongarra,et al.  A Test Matrix Collection for Non-Hermitian Eigenvalue Problems , 1997 .

[10]  Y. Saad.  Numerical Methods for Large Eigenvalue Problems , 2011 .

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

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

[13]  Serkan Gugercin,et al.  Interpolatory Model Reduction of Large-Scale Dynamical Systems , 2010 .

[14]  G. Stewart.  Backward error bounds for approximate Krylov subspaces , 2002 .

[15]  R. Fletcher.  Conjugate gradient methods for indefinite systems , 1976 .

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

[17]  G. Stewart.  Matrix Algorithms, Volume II: Eigensystems , 2001 .

[18]  C. Paige.  Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix , 1976 .

[19]  Zhi-Hao Cao.  On the QMR approach for iterative methods including coupled three-term recurrences for solving nonsymmetric linear systems , 1998 .

[20]  G. Meurant.  The Lanczos and Conjugate Gradient Algorithms: From Theory to Finite Precision Computations , 2006 .

[21]  Zdenek Strakos,et al.  Model reduction using the Vorobyev moment problem , 2009, Numerical Algorithms.

[22]  Valeria Simoncini,et al.  Recent computational developments in Krylov subspace methods for linear systems , 2007, Numer. Linear Algebra Appl..

[23]  C. Lanczos.  Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[24]  Steffen Börm,et al.  Numerical Methods for Eigenvalue Problems , 2012 .

[25]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[26]  B. Parlett,et al.  Semi-duality in the two-sided lanczos algorithm , 1993 .

[27]  Zhaojun Bai,et al.  Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem , 1994 .

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