RD-rational approximations of the matrix exponential
Given a square matrix \(A\) and a vector \(v\), the authors propose two methods for approximating the function \(\exp(-tA) v\). Both methods are based on the Krylov subspace of the matrix \(Z = (I+hA)^{-1}\) applied to \(v\) for some fixed parameter \(h\). If \(V\) denotes an orthonormal basis of this Krylov subspace then the first method (called Arnoldi RD-method) uses the approximation \(V \exp(-\frac t h (H^{-1}-I)) V^T v\), where \(H = V^T A V\). This is shown to be closely related with the polynomial interpolation of the function \(f(z) = \exp(-\frac t h (1/z-1))\) in the eigenvalues of \(H\). The other method (called Arnoldi RF-method) is obtained by using the zeros of Faber polynomials as interpolation points. Some error bounds are provided and numerical experiments compare the two methods with other existing methods for approximating \(\exp(-tA) v\), such as the classical Krylov subspace method and Padé approximations.
scientific article; zbMATH DE number 2153696