Barycentric Lagrange Interpolation Methods for Evaluating Singular Integrals

Abstract

We investigate new, straightforward methods for interpolating and integrating discontinuous single and two-variable real valued functions. The method is based on some modified matrix–vector barycentric Lagrange interpolation formulas. We developed seven rules for the optimum distribution of nodes inside the domain of integration: five for single-valued discontinuous functions, and two rules for the two independent variables of discontinuous functions. We designed these rules to depend on the length of the integration domain and the degree of the interpolant polynomials. Thus, we obtained uniform interpolation and minimum roundoff errors. Based on these rules with the application of the modified matrix–vector barycentric formulas, we easily isolated the singularities of the interpolant integrands and evaluated the corresponding interpolant integral values with super accuracy. The obtained numerical solutions of the five given examples show the high accuracy and efficiency of the presented method compared with the exact solutions and with the cited method.