Error Bounds for Lanczos Approximations of Rational Functions of Matrices
Numerical Validation in Current Hardware Architectures
Extended Krylov subspace for parameter dependent systems
Applied Numerical Mathematics
Shift-Invert Arnoldi Approximation to the Toeplitz Matrix Exponential
SIAM Journal on Scientific Computing
Computing $f(A)b$ via Least Squares Polynomial Approximations
SIAM Journal on Scientific Computing
Computation of matrix functions with deflated restarting
Journal of Computational and Applied Mathematics
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Building upon earlier work by Golub, Meurant, Strakoš, and Tichý, we derive new a posteriori error bounds for Krylov subspace approximations to $f(A)b$, the action of a function $f$ of a real symmetric or complex Hermitian matrix $A$ on a vector $b$. To this purpose we assume that a rational function in partial fraction expansion form is used to approximate $f$, and the Krylov subspace approximations are obtained as linear combinations of Galerkin approximations to the individual terms in the partial fraction expansion. Our error estimates come at very low computational cost. In certain important special situations they can be shown to actually be lower bounds of the error. Our numerical results include experiments with the matrix exponential, as used in exponential integrators, and with the matrix sign function, as used in lattice quantum chromodynamics simulations, and demonstrate the accuracy of the estimates. The use of our error estimates within acceleration procedures is also discussed.