Interpolació i quadratura en dues variables

Abstract

[en] Numerical integration, also known as quadrature, belongs to the group of problems of numerical analysis. The quadrature study has been extensively developed for functions of one dimension. Although it has many applications in various areas of science, the numerical calculation of integrals in higher dimensions has not been so much worked, mainly because of the difficulties involved in the fact that there are many more integration regions than in the 1-dimensional case. In this study it is developed the quadrature in two variables, explaining with detail some of the methods that can be used, and presenting practical results to support theoretical concepts. Specifically, the quadrature product formulas are developed, using the results of dimension 1 to arrive at valid 2-dimensional methods. Mainly, Simpson’s quadrature method and the 1-dimensional Trapezoidal rule are used to extend it to integrals on rectangular regions in the plane. Finally, the concept of adaptive quadrature is introduced and worked, which aims to reduce the cost and improve the computational efficiency of the product formulas, taking into account the nature of the function in the different zones of the integration region.