Efficient Gauss-related quadrature for two classes of logarithmic weight functions

Integrals with logarithmic singularities are often difficult to evaluate by numerical methods. In this work, a quadrature method is developed that allows the exact evaluation (up to machine accuracy) of integrals of polynomials with two general types of logarithmic weights. The total work for the determination of N nodes and points of the quadrature method is O(N-2). Subsequently, integrals can be evaluated with O(N) operations and function evaluations, so the quadrature is efficient. This quadrature method can then be used to generate the nonclassical orthogonal polynomials for weight functions containing logarithms and obtain Gauss and Gauss-related quadratures for these weights. Two algorithms for each of the two types of logarithmic weights that incorporate these methods are given in this paper.