Gaussian integration with rescaling of abscissas and weights

An algorithm for integration of the polynomial functions with a variable weight is considered. It provides an extension of the Gaussian integration, with appropriate scaling of the abscissas and weights. In a first step, orthogonal polynomials are computed for a fixed a=1a=1. Then, using approximate scaling, the initial guess is constructed for a≠1a≠1. Finally, numerical values of the abscissas and weights are refined, solving polynomial system using Newton–Raphson method. The final form of the algorithm provides good alternative to usually adopted interval splitting, automatically avoiding problems with limiting values of parameter present in the weight function. Construction of the method requires arbitrary precision arithmetic and special functions, polylogarithms in particular. The final form of the algorithm can be coded using machine precision floating point numbers and standard mathematical library.

► Numerical integration.
► Non-classic orthogonal polynomials.
► Gaussian quadratures.
► Polynomial systems: solving, scaling and symmetries.