Residual, restarting and Richardson iteration for the matrix exponential

Mike A. Bochev, Volker Grimm, Marlis Hochbruck

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    45 Citations (Scopus)
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    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 [SIAM J. Sci. Comput., 19 (1998), pp. 38–54] and interpret the sought-after vector as the value of a vector function satisfying the linear system of ordinary differential equations (ODEs) 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 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.
    Original languageEnglish
    Pages (from-to)A1376-A1397
    Number of pages22
    JournalSIAM journal on scientific computing
    Issue number3
    Publication statusPublished - 2013


    • MSC-65F60
    • Backward stability
    • MSC-65N22
    • MSC-65F10
    • MSC-65F30
    • MSC-65L05
    • Residual
    • Stopping criterion
    • Krylov subspace methods
    • Matrix cosine
    • Matrix exponential
    • Restarting
    • Richardson iteration
    • Chebyshev polynomials


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