Deflated Restarting for Matrix Functions

We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is to ultimately deflate a specific invariant subspace of the matrix which most impedes the convergence of the restarted approximation process. An approximation to the subspace to be deflated is successively refined in the course of the underlying restarted Arnoldi process by extracting Ritz vectors and using those closest to the spectral region of interest as exact shifts. The approximation is constructed with the help of a generalization of Krylov decompositions to linearly dependent vectors. A description of the restarted process as a successive interpolation scheme at Ritz values is given in which the exact shifts are replaced with improved approximations of eigenvalues in each restart cycle. Numerical experiments demonstrate the efficacy of the approach.

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