Topics in matrix analysis
New computer methods for global optimization
New computer methods for global optimization
Two polynomial methods of calculating functions of symmetric matrices
USSR Computational Mathematics and Mathematical Physics
Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
Efficient solution of parabolic equations by Krylov approximation methods
SIAM Journal on Scientific and Statistical Computing
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Preconditioning Lanczos Approximations to the Matrix Exponential
SIAM Journal on Scientific Computing
Analysis of Projection Methods for Rational Function Approximation to the Matrix Exponential
SIAM Journal on Numerical Analysis
Stopping Criteria for Rational Matrix Functions of Hermitian and Symmetric Matrices
SIAM Journal on Scientific Computing
Acceleration Techniques for Approximating the Matrix Exponential Operator
SIAM Journal on Matrix Analysis and Applications
Exponential Runge--Kutta methods for parabolic problems
Applied Numerical Mathematics
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Having good estimates or even bounds for the error in computing approximations to expressions of the form f(A)v is very important in practical applications. In this paper we consider the case that A is Hermitian and that f is a rational function. We assume that the Lanczos method is used to compute approximations for f(A)v and we show how to obtain a posteriori upper and lower bounds on the ℓ2-norm of the approximation error. These bounds are computed by minimizing and maximizing a rational function whose coefficients depend on the iteration step. We use global optimization based on interval arithmetic to obtain these bounds and include a number of experimental results illustrating the quality of the error estimates.