Numerical quadrature and nonlinear sequence transformations; unified rules for efficient computation of integrals with algebraic and logarithmic endpoint singularities

Abstract

Some nonlinear transformations for accelerating the convergence of infinite sequences due to Levin are reviewed, and new results of practical importance in applications are given. Using these results, the transformations of Levin are modified and used to obtain new numerical integration formulas for weight functions with algebraic and logarithmic endpoint singularities, which are simpler to compute and practically as efficient as the corresponding Gaussian formulas. They also have the additional advantage that different weight functions of a certain type can have the same set of abscissas associated with them. It is shown that the formulas obtained are of interpolatory type. Furthermore, for some cases it is proved that the abscissas are in the interval of integration, although numerical results indicate that this is so in all cases and that the weights are all positive. Several numerical examples that illustrate the high accuracy and convenience of the new formulas are appended.