Pub. online:22 Nov 2023Type:Research ArticleOpen Access
Journal:Informatica
Volume 35, Issue 1 (2024), pp. 21–46
Abstract
In this paper, the numerical algorithms for calculating the values of the left- and right-sided Riemann–Liouville fractional integrals and the Riesz fractional integral using spline interpolation techniques are derived. The linear, quadratic and three variants of cubic splines are taken into account. The estimation of errors using analytical methods are derived. We show four examples of numerical evaluation of the mentioned fractional integrals and determine the experimental rate of convergence for each derived algorithm. The high-precision calculations are executed using the 128-bit floating-point numbers and arithmetic routines.
Journal:Informatica
Volume 4, Issues 1-2 (1993), pp. 227–249
Abstract
This work was stimulated by investigations on Markow Renewal Processes. For finding analytic solutions (to compute the probabilities of certain states of the system) multivariate Laplace transforms can be used. Tables with correspondences of function and their transforms very rarely help to solve such problems.
In Chapter I number theoretical numerics are applied to compute the original function of a multivariate Laplace transform given. Starting with the complex multivariate inversion theorem the domain of integration is mapped onto the s-dimensional unit cube Gs. Using a periodization of the integrand new results concerning the vanishing of the multivariate Laplace transform in regard of the modified numerical inversion formula are shown.
In Chapter II two implementations are discussed: A method to implement a Manager-Worker Process (MWP) to reduce the idle times of the processors is presented and the tasks of the Manager and the Workers are defined. The numerical inversion using this method with STRAND88 has been implemented on a heterogenous workstation net. The MWP provided a good load balancing. Another implementation with C-LINDA has been done on a Shared Memory MIMD system. We also implemented a kind of MWP. Numerical experiments have shown that the decomposition of the problem is sufficiently.