Extended Gaussian quadratures for functions with an end-point singularity of logarithmic type

The extended Gaussian quadrature rules are shown to be an efficient tool for numerical integration of wide class of functions with singularities of logarithmic type. The quadratures are exact for the functions pol1n−1(x) + lnx pol2n−1(x) , where pol1n−1(x) and pol2n−1(x) are two arbitrary polynomials of degree n−1 and n is the order of the quadrature formula. We present an implementation of numerical algorithm that calculates the nodes and the weights of the quadrature formulas, provide Fortran code for numerical integration, and test the performance of different kinds of Gaussian quadratures for functions with logarithmic singularities.