On convergence of Krylov subspace approximations of time-invariant self-adjoint dynamical systems

Abstract

We extend the rational Krylov subspace algorithm from the computation of the action of the matrix exponential to the solution of stable dynamical systemsA˜ddtu(t)=b(t),u|t<0=0,A˜ddt=∑i=0mAiddt+sIi,where m∈N∪{∞}, Ai=Ai∗∈RN×N,s⩽0, and u(t),b(t)∈RN,b|t<0=0 (not assuming that evolution of b(t) is described by a low-dimensional subspace of RN). We show that the reduced equation is stable and derive an a priori error bound via rational approximation of the exponential on the boundary of the nonlinear numerical range of A˜. We also describe a simple and easily computable external bound of this numerical range. The obtained results are applied to the infinite order problem arising in the solution of the dispersive Maxwell’s system.