Extension of numerical quadrature formulae to cater for end point singular behaviours over finite intervals

Abstract

The N-point Gaussian quadrature method is generalised to cater for various possible singular behaviours at the end points of the interval of integration at the expense of being algebraically exact for a polynomial of lower order than usual. Weights and abscissae are chosen to exactly integrate an integrand which is the sum of the singular functions and an arbitrary polynomial. This allows us to cater for several different end-point singularities in the same quadrature formula and in this way differs from published quadratures where a singular behaviour is incorporated in a weight function that multiplies an arbitrary polynomial. We present tables of weights and abscissae that cater for (i) logarithmic end point singularities and (ii) logarithmic plus inverse square root singular behaviours. Also a 10-point quadrature is presented that exactly caters for log(x), x^-1/4,x^-1/2,x^-3/4 singular behaviours and is recommended for programmable calculator use. Finally a brief comparison study of the various (10-point) quadratures herein considered is made.