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The approximation of $e^{tA}B$ where $A$ is a large sparse matrix and $B$ arectangular matrix is the key ingredient in many scientific and engineeringcomputations. A powerful tool to manage the matrix exponential function is toresort to a suitable rational approximation such as the Carath$\acute{\rme}$odory-Fej$\acute{\rm e}$r approximation, whose core reduces to solve shiftedlinear systems with multiple right-hand sides. The restarted and shifted blockFOM algorithm is a commonly used technique for this problem. However,determining good preconditioners for shifted systems that preserve the originalstructure is a difficult task. In this paper, we propose a new preconditionerfor the restarted and shifted block FOM algorithm. The key is that the absolutevalues of the poles of the Carath$\acute{\rm e}$odory-Fej$\acute{\rm e}$rapproximation are medium sized and can be much smaller than the norm of thematrix in question. The advantages of the proposed strategy are that we canprecondition all the shifted linear systems simultaneously, and preserve theoriginal structure of the shifted linear systems after restarting. Theoreticalresults are provided to show the rationality of our preconditioning strategy.Applications of the new approach to Toeplitz matrix exponential problem arealso discussed. Numerical experiments illustrate the superiority of the newalgorithm over many state-of-the-art algorithms for matrix exponential.

Preconditioning the Restarted and Shifted Block FOM Algorithm for Matrix Exponential Computation