Error Bounds for the Krylov Subspace Methods for Computations of Matrix Exponentials

Error Bounds for the Krylov Subspace Methods for Computations of Matrix Exponentials

In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing e−τAv for a given τ > 0 and v ∈ Cn, where A is a large sparse nonHermitian matrix. The a priori error bounds relate the convergence to λmin( A+A∗ 2 ), λmax( A+A∗ 2 ) (the smallest and the largest eigenvalue of the Hermitian part of A), and |λmax(A−A 2 )| (the largest eigenvalue in ...

Error Bounds for the Lanczos Methods for Approximating Matrix Exponentials

In this paper, we present new error bounds for the Lanczos method and the shift-andinvert Lanczos method for computing e−τAv for a large sparse symmetric positive semidefinite matrix A. Compared with the existing error analysis for these methods, our bounds relate the convergence to the condition numbers of the matrix that generates the Krylov subspace. In particular, we show that the Lanczos m...

Krylov Subspace Methods for Tensor Computations

A couple of generalizations of matrix Krylov subspace methods to tensors are presented. It is shown that a particular variant can be interpreted as a Krylov factorization of the tensor. A generalization to tensors of the Krylov-Schur method for computing matrix eigenvalues is proposed. The methods are intended for the computation of lowrank approximations of large and sparse tensors. A few nume...

Exploiting Structure in Krylov Subspace Methods for the WilsonFermion Matrix

Krylov subspace methods for the Dirac equation

The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos alg...