Modified newton–cotes formulae for numerical quadrature of oscillatory integrals with two independent variable frequencies

Abstract

It is shown how a function can be approximated in a unique way by an interpolation function f_n (x) which is a linear combination of cos kx, sin kx, cos k′x, sin k′xand a polynomial of {n– 4)th degree, such that f_n coincides with f(x) in (n + 1) equidistant points, and whereby kand k′are two arbitrary real, purely imaginary or complex conjugate parameters. Several equivalent expressions of the interpolation function f_n are given, and also the error term is derived in closed form. With this type of interpolation a set of modified Newton–Cotes quadrature rules is established and the total truncation error associated with these rules is discussed. Subsequently, the five-point quadrature rule is treated in full detail and several formulae are derived which facilitate numerical computation. Finally, the proposed formulae are implemented for certain illustrative numerical examples. In particular, a technique is established for attributing values to the parameters k and k′ such that the total truncation error is minimized.

Keywords:

• Interpolation function
• Numerical quadrature
• Newton-Cotes rules

C.R. Categories:

• G.1.1
• G.1.4

^∗Research Director at the National Fund for Scientific Research (N.F.W.O. Belgium)

^∗Research Director at the National Fund for Scientific Research (N.F.W.O. Belgium)

Notes

^∗Research Director at the National Fund for Scientific Research (N.F.W.O. Belgium)