Journal of Approximation Software

Preface of JAS-01-01-2024

2024-07-23T18:10:57+02:00

We have the pleasure to present the first issue of the Journal of Approximation Software. It publishes in open-access mode well-structured accompanying articles to open-source software, on all aspects of approximation theory and applications in its broadest sense.

The publisher is University of Torino (Italy) by its open-access platform SIRIO@UniTO.

Implementations can use the most common languages and environments like MATLAB, Python, C, C++, among others. The codes are posted by the authors on stable public platforms with a clear software versioning policy, such as GitHub.

There will be one general issue per year (with articles published continuously by acceptance date) and possible thematic special issues including conference proceedings.

This first issue collects a few regular articles of colleagues and researchers, who work in different fields of approximation implementing numerical algorithms by means of the MATLAB software. The issue includes a variety of approximation topics involving exponential integrators, ordinary differential equations, quadrature of oscillating functions, computation of derivatives and QuasiMonteCarlo integration.

We sincerely thank the authors of the papers and the anonymous referees helping us to carefully review the papers included in this first issue of the first volume of JAS.

Direction splitting of ϕ-functions in exponential integrators for d-dimensional problems in Kronecker form

2024-07-23T18:15:50+02:00

In this manuscript, we propose an efficient, practical and easy-to-implement way to approximate actions of ϕ-functions for matrices with d-dimensional Kronecker sum structure in the context of exponential integrators up to second order. The method is based on a direction splitting of the involved matrix functions, which lets us exploit the highly efficient level 3 BLAS for the actual computation of the required actions in a µ-mode fashion. The approach has been successfully tested on two- and three-dimensional problems with various exponential integrators, resulting in a consistent speedup with respect to a technique designed to approximate actions of ϕ-functions for Kronecker sums.

Qsurf: compressed QMC integration on parametric surfaces

2024-07-23T18:17:22+02:00

We discuss a “bottom-up” algorithm for Tchakaloff-like compression of QMC (QuasiMonteCarlo) integration on surfaces that admit an analytic parametrization. The key tools are Davis-Wilhelmsen theorem on the so-called “Tchakaloff sets” for positive linear functionals on polynomial spaces, and Lawson-Hanson algorithm for NNLS. This algorithm shows remarkable speed-ups with respect to Caratheodory-like subsampling, since it is able to work with much smaller matrices. We provide the corresponding Matlab code Qsurf, together with integration tests on regions of different surfaces such as sphere, torus, and a smooth Cartesian graph.

NDED - Numerical derivatives from equispaced data

2024-07-23T18:17:57+02:00

Procedure NDED computes the numerical derivatives of order ν from equispaced data. This is based on the iterated application of a spectral algorithm for the computation of the first order derivative. A preliminary test of the procedure gives satisfactory results.

A MATLAB implementation of TASE-RK methods

2024-07-23T18:18:33+02:00

In this paper, we analyze theoretical and implementation aspects of Time-Accurate and highly-Stable Explicit Runge-Kutta (TASE-RK) methods, which have been recently introduced by Bassenne et al. (2021) [5], for the numerical solution of stiff Initial Value Problems (IVPs). These methods are obtained by combining explicit RK schemes with suitable matrix operators, called TASE operators, involving in their expression a matrix J related to the Jacobian of the differential problem to be solved. By analyzing the formulation and order conditions of TASE-RK methods, we show that they can be interpreted as particular linearly implicit RK schemes, and that their consistency properties are independent of the choice of J. Using this information, we propose a MATLAB implementation of TASE-RK methods, which makes use of matrix factorizations and allows setting J according to user preferences.

Numerical quadrature for integrals involving oscillating functions

2024-07-23T18:19:13+02:00

This paper deals with the construction of a coupled Gaussian rule for weight functions involving powers, exponentials and trigonometric functions. Starting from a three term recursion for the moments, nodes and weights are computed by using the Chebyshev algorithm together with the Golub and Welsch method. An a posteriori approximation of the quadrature error by means of the generalized averaged Gaussian rules is also considered. Several numerical examples are provided.