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      Residual, restarting and Richardson iteration for the matrix exponential, revised

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          Abstract

          A well-known problem in computing some matrix functions iteratively is the lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Suppose the matrix exponential of a given matrix times a given vector has to be computed. We develop the approach of Druskin, Greenbaum and Knizhnerman (1998) and interpret the sought-after vector as the value of a vector function satisfying the linear system of ordinary differential equations (ODE) whose coefficients form the given matrix. The residual is then defined with respect to the initial-value problem for this ODE system. The residual introduced in this way can be seen as a backward error. We show how the residual can be computed efficiently within several iterative methods for the matrix exponential. This completely resolves the question of reliable stopping criteria for these methods. Further, we show that the residual concept can be used to construct new residual-based iterative methods. In particular, a variant of the Richardson method for the new residual appears to provide an efficient way to restart Krylov subspace methods for evaluating the matrix exponential.

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          Expokit: a software package for computing matrix exponentials

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            Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator

            Y Saad (1992)
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              Symmetry-preserving discretization of turbulent flow

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                Author and article information

                Journal
                23 December 2011
                Article
                1112.5670
                2db1009d-3c15-4343-9994-7a7c46e0fc4c

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                65F60, 65F10, 65F30, 65N22, 65L05
                24 pages
                math.NA cs.CE physics.comp-ph

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