On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

Abstract

With existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the formwhere f is a sufficiently smooth function on [0, 1]. Gautschi has developed software (as scripts in Matlab) for computing these integrals for the special case r = ω = 1. In this paper, an algorithm (as a Mathematica program) is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range. Numerical examples are given of testing the performance of the algorithm/program.

Error bounds for approximation in Chebyshev points

“…(see [15]). Before we further discuss the error estimate, we first take f (x) = 1 1 + 16x 2 as an example to illustrate the convergence of the quadrature error in the Chebyshev points and equispaced points for the highly oscillatory integrals, respectively.…”

“…, ω sin(r π) − iω cos(r π) (r < 0) (see [15]). In the case r > 0: From Lemma 3.2 for x ∈ [0, 1] it follows that…”

An efficient adaptive Levin-type method for highly oscillatory integrals

“…where δ = δ(a, b) = min {|ωg (a)|, |ωg (b)|} as before and N 0 = n/2 ; otherwise divide the integration interval into subintervals and use either the Clenshaw-Curtis points (9) or the points (10) for…”

“…Unfortunately, it requires that the integrand be analytic in a simply connected and sufficiently large region Ω ⊂ C containing [a, b]. Moreover it requires the calculation of inverse of g. For some of the methods for highly oscillatory integrals with infinite integration interval or with singular integrand, see, e.g., [4,7,8,9,10,11,15].…”

Numerical Evaluation Of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity

“…The precision is worse than with Algorithm 1 by approximately one digit. I have not looked into advanced schemes for this type of oscillatory integrals [7,11] or Sidi's generalized methods of extrapolation.…”