Computing the Hilbert transform on the real line
Mathematics of Computation
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Exponential Integrators for Large Systems of Differential Equations
SIAM Journal on Scientific Computing
Numerical Inversion of Laplace Transforms Using Laguerre Functions
Journal of the ACM (JACM)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
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We consider the approximation of semigroups $e^{\tau A}$ and of the functions $\varphi_j(\tau A)$ that appear in exponential integrators by resolvent series. The interesting fact is that the resolvent series expresses the operator functions $e^{\tau A}$ and $\varphi_j(\tau A)$, respectively, in efficiently computable terms. This is important for semigroups, where the new approximation is different from well-known approximations by rational functions, as well as for the application of exponential integrators, which are currently of high interest and which are usually studied in a semigroup setting on Banach spaces. The approximation of the operator functions $\varphi_j(\tau A)$ in a general, strongly continuous semigroup setting has not been discussed in the literature so far, but this is crucial for an application of these integrators with unbounded operators or bounded operators (like discretization matrices) with large norm and eigenvalues somewhere in the left half plane.