Calculation of Gaussian quadratures with multiple nodes

Abstract

A new algorithm for constructing quadrature formulas with multiple Gaussian nodes in the presence of certain fixed nodes is presented. It can be used in construction for measures with the bounded and infinite supports. The algorithm includes a simple and fast method for finding elements of the Jacobian matrix in an iterative process with quadratic convergence, which is based on using the fundamental three-term recurrence relation for the corresponding orthogonal polynomials with respect to a given measure. Numerical experiments with respect to several different measures are also included. Finally, a simple method for calculating weight coefficients (Cotes numbers of higher order) of Gaussian and prescribed nodes is presented. Numerical examples are included.