Error estimates for Krylov subspace approximations 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 ...

Comparing Leja and Krylov Approximations of Large Scale Matrix Exponentials

We have implemented a numerical code (ReLPM, Real Leja Points Method) for polynomial interpolation of the matrix exponential propagators exp (∆tA)v and φ(∆tA)v, φ(z) = (exp (z) − 1)/z. The ReLPM code is tested and compared with Krylov-based routines, on large scale sparse matrices arising from the spatial discretization of 2D and 3D advection-diffusion equations.

On Krylov Subspace Approximations to Thematrix Exponential

Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator

In this note we present a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation exp(A)v and establish a priori and a posteriori error estimates. Several such approximations are considered. The main idea of these techniques is to approximately project the exponential operator onto a small Krylov subspace and carry out the resulting small exponential matr...

Sharp Ritz Value Estimates for Restarted Krylov Subspace Iterations

Gradient iterations for the Rayleigh quotient are elemental methods for computing the smallest eigenvalues of a pair of symmetric and positive definite matrices. A considerable convergence acceleration can be achieved by preconditioning and by computing Rayleigh-Ritz approximations from subspaces of increasing dimensions. An example of the resulting Krylov subspace eigensolvers is the generaliz...