Integració numèrica: fórmules interpolatòries, gaussianes, el mètode doble-exponencial i exemples

Abstract

[en] The integral over a bounded interval $[a, b]$ of a continuous $f$ function can be numerically approximated using interpolation or Gaussian quadrature formulas, which evaluate the function $f$ in $m + 1$ abscissas. The choice of abscissas is essential in Gaussian integration to obtain exact formulas for polynomials of degree less than or equal to $2m + 1$, improving the accuracy of interpolation formulas. From a numerical point of view, quadrature formulas that provide a small error with few evaluations of $f$ are needed. One of these formulas is the one provided by the double exponential method, which consist of using a suitable transformation that maps the endpoints of the interval of integration to infinity, in order to obtain a double exponential decay of the integrant, and then apply the trapezoidal formula. We will see that this method allows to approximate converging improper integrals with singularities at the endpoints and that it can be adapted to approximate oscillatory integrals with different types of decay. Oscillatory integrals with a fast decay of the integrand are naturally found in dynamical systems when the Melnikov integral is considered to measure the splitting of separatrices. Throughout the work, some examples are analyzed and compared with numerical implementations using Pari/GP software.