Fast methods for the computation of the Mittag-Leffler function

In this chapter we discuss the problem of the fast and reliable computation of the Mittag-Leffler function. We start from the series representation, by which the function is commonly defined, to illustrate how it turns out to be not suitable for computation in most of the cases; then we present an alternative approach based on the numerical inversion of the Laplace transform. In particular, we describe a technique, known as the optimal parabolic contour, in which the inversion of the Laplace transform is performed by applying the trapezoidal rule along a parabolic contour in the complex plane; the contour and the integration parameters are chosen, on the basis of the error analysis and the location of the singularities of the Laplace transform, with the aim of achieving a target accuracy (which can be very close to machine precision) with a substantially low computational effort. Applications to the evaluation of derivatives of the Mittag-Leffler function and to matrix arguments are also discussed.