Dual Mixed Gaussian Quadrature Based Adaptive Scheme for Analytic Functions

Abstract

An efficient adaptive scheme based on a dual mixed quadrature rule of precision eleven for approximate evaluation of line integral of analytic functions has been constructed. At first, the precision of Gauss-Legendre four point transformed rule is enhanced by using Richardson extrapolation. A suitable convex combination of the resulting rule and the Gauss-Legendre five point rule further enhances the precision producing a new mixed quadrature rule . This mixed rule is termed as dual mixed Gaussian quadrature rule as it acquires a very high precision eleven using Gaussian quadrature rule in two steps. An adaptive quadrature scheme is designed .Some test integrals having analytic function integrands have been evaluated using the dual mixed rule and its constituent rules in non- adaptive mode. The same set of test integrals have been evaluated using those rules as base rules in the adaptive scheme. The dual mixed rule based adaptive scheme is found to be most effective.