Numerical integration with highly oscillating weight functions

Numerical integration of highly–oscillating functions

By a highly–oscillating function we mean one with large number of local maxima and minima over some interval. The computation of integrals of highly–oscillating functions is one of the most important issues in numerical analysis since such integrals abound in applications in many branches of mathematics as well as in other sciences, e.g., quantum physics, fluid mechanics, electromagnetics, etc....

Moment-free numerical integration of highly oscillatory functions

The aim of this paper is to derive new methods for numerically approximating the integral of a highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Using a method developed by Levin as a point of departure, we construct a new method that utilizes the same information as the Filon-type method, and obtains the same asymptot...

Numerical Integration with Complex Jacobi Weight Function

In this paper we study the numerical integration on (−1, 1) with respect to the Jacobi weight function (1−x)(1+x), where α and β are complex parameters. The problem arises in some applications of computational models in quantum mechanics. We discuss two methods for integration. One is suitable for integration of analytic functions and the other is applicable to the general Riemann integrable fu...

A Numerical Integration Method by Using Generalized Series of Functions

Numerical indefinite integration of functions with singularities

We derive an indefinite quadrature formula, based on a theorem of Ganelius, for Hp functions, for p > 1, over the interval (−1, 1). The main factor in the error of our indefinite quadrature formula is O(e−π √ ), with 2N nodes and 1 p + 1 q = 1. The convergence rate of our formula is better than that of the Stenger-type formulas by a factor of √ 2 in the constant of the exponential. We conjectur...