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

Abstract

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)+lnxpol2n−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 a Fortran code for numerical integration, and test the performance of different kinds of Gaussian quadratures for functions with logarithmic singularities. Program summaryProgram title: GAUSEXTCatalogue identifier: AETP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETP_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2535No. of bytes in distributed program, including test data, etc.: 39963Distribution format: tar.gzProgramming language: Mathematica, Fortran.Computer: PCs or higher performance computers.Operating system: Linux, Windows, MacOS.RAM: Kilobytes.Classification: 4.11.Nature of problem:Quadrature formulas for numerical integration, effective for a wide class of functions with end-point singularities of logarithmic type.Solution method:The method of solution is based on the algorithm developed in Ref. [1] with some modifications.Running time:Milliseconds to minutes.