Computing the coefficients of a recurrence formula for numerical integration by moments and modified moments (Q1318409)

The authors consider the evaluation of the class of integrals \(\int^ 1_{-1} e^{-\alpha x} f(x)dx\), where \(\alpha\) is a positive number and the function \(f\) is known only approximately on a discrete set of points. They use a Gaussian quadrature formula in which the nodes and weights have to be computed via a family of monic orthogonal polynomials with respect to the weight function through the three-term recurrence relation \[ P_{k+1}(x)= (x+ B_{k+1}) P_ k(x)- C_{k+1} P_{k-1}(x). \] The thrust of this paper is a careful evaluation of the coefficients to guarantee a good precision through a Mathematica program. A comparison between various methods, starting from moments and modified moments is shown. Numerical results are also presented.