Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.

In this paper we investigate some well-established and more recent methods that aim at approximating the vector $\exp(A)v$ when A is a large symmetric negative semidefinite matrix, by efficiently combining subspace projections and spectral transformations. We show that some recently developed acceleration procedures may be restated as preconditioning techniques for the partial fraction expansion form of an approximating rational function. These new results allow us to devise a priori strategies to select the associated acceleration parameters; theoretical and numerical results are shown to justify these choices. Moreover, we provide a performance evaluation among several numerical approaches to approximate the action of the exponential of large matrices. Our numerical experiments provide a new, and in some cases, unexpected picture of the actual behavior of the discussed methods.