Filon quadrature
In numerical analysis, Filon quadrature or Filon's method is a technique for numerical integration of oscillatory integrals. It is named after English mathematician Louis Napoleon George Filon, who first described the method in 1934.
Description
The method is applied to oscillatory definite integrals in the form:where is a relatively slowly-varying function and is either sine or cosine or a complex exponential that causes the rapid oscillation of the integrand, particularly for high frequencies. In Filon quadrature, the is divided into subintervals of length, which are then interpolated by parabolas. Since each subinterval is now converted into a Fourier integral of quadratic polynomials, these can be evaluated in closed-form by integration by parts. For the case of, the integration formula is given as:
where
Explicit Filon integration formulas for sine and complex exponential functions can be derived similarly. The formulas above fail for small values due to catastrophic cancellation; Taylor series approximations must be in such cases to mitigate numerical errors, with being recommended as a possible switchover point for 44-bit mantissa.
Modifications, extensions and generalizations of Filon quadrature have been reported in numerical analysis and applied mathematics literature; these are known as Filon-type integration methods. These include Filon-trapezoidal and Filon–Clenshaw–Curtis methods.