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Exponential time integrators are well‐established discretization methods for time semilinear systems of ordinary differential equations. These methods use φ−functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the φ−function action over vectors for two‐dimensional (2D) matrices expressed as a Kronecker sum. For that, we present an auxiliary exponential‐related matrix function that we express using Kronecker products of one‐dimensional matrices. We exploit state‐of‐the‐art implementations of φ−functions to compute this auxiliary function's action and then recover the original φ−action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen–Cahn equation. [ABSTRACT FROM AUTHOR]

Summary: A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, the numerical evaluation of matrix functions expressed in partial fraction form is addressed. The shift-and-invert Krylov method is analyzed, with special attention to error estimates. Such estimates give insights into the selection of the shift parameter and lead to a simple and effective restart procedure. Applications to the class of Mittag-Leffler functions are presented. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper deals with the application of the restricted-denominator rational Krylov method, recently discussed in (BIT 2004; 44(3):595–615; SIAM J. Sci. Comput. 2005; 27:1438–1457), to the computation of the action of the so-called ϕ-functions, which play a fundamental role in several modern exponential integrators. The analysis here presented is devoted in particular to the construction of error estimates of easy practical use. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]