Computing Approximate (Symmetric Block) Rational Krylov Subspaces without Explicit Inversion

It has been shown that approximate extended Krylov subspaces can be computed –under certain assumptions– without any explicit inversion or system solves. Instead the necessary products A−1v are obtained in an implicit way retrieved from an enlarged Krylov subspace. In this paper this approach is generalized to rational Krylov subspaces, which contain besides poles at infinite and zero also finite non-zero poles. Also an adaption of the algorithm to the block and the symmetric case is presented. For all variants of the algorithm numerical experiments underpin the power of the new approach. Rational Krylov subspaces can be used, e.g., to approximate matrix functions or the solutions of matrix equations.