This paper addresses the problem of the numerical computation of generalized Mittag–Leffler functions with two parameters, with applications in fractional calculus. The inversion of their Laplace transform is an effective tool in this direction; however, the choice of the integration contour is crucial. Here parabolic contours are investigated and combined with quadrature rules for the numerical integration. An in-depth error analysis is carried out to select suitable contour’s parameters, depending on the parameters of the Mittag– Leffler function, in order to achieve any fixed accuracy.We present numerical experiments

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.

In this paper we consider the numerical computation of the matrix pth root of stochastic matrices. In particular a collection of theoretical results concerning the pth root of stochastic matrices is reported and some numerical methods are described. The aim of the paper is to highlight the properties of such methods when they are applied to numerically compute the pth root of a stochastic matrix when it is expected to be stochastic too.

Transition matrices arise in a wide class of engineering problems especially in mathematical models using Markov chains. Usually they describe the transition probabilities of a vector state at time $t$ to the same state vector at time $t+Delta t$, where $Delta t$ is the shortest period over which a transition matrix can be estimated. If a short term transition matrix is needed it can be obtained by computing a $p$th root. For example in risk management of portfolio the company's credit ratings are recorded yearly, thus to define, at the end of the year, a transition matrix with all the recorded information. However, investment horizon is shorter than a year, thus to require the computation of the matrix $p$th root. In this paper we consider the numerical computation of the $p$th root of a transition matrix. The aim of the paper is to highlight the properties of some numerical methods preserving the geometric peculiarities of the pth root of a transition matrix.