Abstract:

The main aim of this project is to use and implement our very recent computational methods, and ones that will be developed within the project, for numerical integration, discuss related applications and provide appropriate software. Central formulas for numerical integration are the famous Gaussian quadratures, introduced by C.F. Gauss more than 200 years ago. In the mid-1960s, Kronrod introduced new quadratures (often called the “20th century quadratures”) used to practically estimate the error in the Gaussian quadratures, but they do not always exist. An alternative to these are the averaged Gaussian quadratures introduced by D. Laurie (Math. Comp. (AMS, 1996)). M. Spalević in his Math. Comp. (2007) paper proposed an optimal variant thereof, along with a numerically effective method of construction. Recently, L. Reichel and M. Spalević [Appl. Numer. Math. (Elsevier, 2021), J. Comput. Appl. Math. (Elsevier, 2022)] proposed more effective construction of the averaged Gaussian quadrature rules based on algebraic polynomials. The same will be done also for quadratures based on trigonometric polynomials. The method will be used in estimating the error in Gaussian formulas, as well as in the approximation of matrix functions and functionals, in solving integral equations numerically, in the summing of number series, etc. We expect to publish 32 papers in leading scientific journals in math and appl math, and 4 monographs with renowned publishers. Some of our methods, in particular relative to the averaged Gaussian quadratures, will hopefully open a new direction in developing the theory of Gaussian rules and become standard both in the research of numerical integration and in teaching materials. Applications of these results are expected to be found in many areas of science and engineering. The project will be realized by 10 participants, helped by 2 external collaborators. This research should significantly increase the scientific potential of the project team.