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.^[1]

The method is applied to oscillatory definite integrals in the form:

${\displaystyle \int _{a}^{b}f(x)g(x)dx}$

where ${\textstyle f(x)}$ is a relatively slowly-varying function and ${\textstyle g(x)}$ 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 ${\textstyle f(x)}$ is divided into ${\textstyle 2N}$ subintervals of length ${\textstyle h}$, 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 ${\textstyle g(x)=\cos(kx)}$, the integration formula is given as:^[1][2]

${\displaystyle \int _{a}^{b}f(x)\cos(kx)dx\approx h(\alpha \left[f(b)\sin(kb)-f(a)\sin(ka)\right]+\beta C_{2n}+\gamma C_{2n-1})}$

where

${\displaystyle \alpha =\left(\theta ^{2}+\theta \sin(\theta )\cos(\theta )-2\sin ^{2}(\theta )\right)/\theta ^{3}}$

${\displaystyle \beta =2\left[\theta (1+\cos ^{2}(\theta ))-2\sin(\theta )\cos(\theta )\right]/\theta ^{3}}$

${\displaystyle \gamma =4(\sin(\theta )-\theta \cos(\theta ))/\theta ^{3}}$

${\displaystyle C_{2n}={\frac {1}{2}}f(a)\cos(ka)+f(a+2h)\cos(k(a+2h))+f(a+4h)\cos(k(a+4h))+\ldots +{\frac {1}{2}}f(b)\cos(kb)}$

${\displaystyle C_{2n-1}=f(a+h)\cos(k(a+h))+f(a+3h)\cos(k(a+3h))+\ldots +f(b-h)\cos(k(b-h))}$

${\displaystyle \theta =kh}$

Explicit Filon integration formulas for sine and complex exponential functions can be derived similarly.^[2] The formulas above fail for small ${\textstyle \theta }$ values due to catastrophic cancellation;^[3] Taylor series approximations must be in such cases to mitigate numerical errors, with ${\textstyle \theta =1/6}$ being recommended as a possible switchover point for 44-bit mantissa.^[2]

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.^[4][5] These include Filon-trapezoidal^[2] and Filon–Clenshaw–Curtis methods.^[6]

Filon quadrature is widely used in physics and engineering for robust computation of Fourier-type integrals. Applications include evaluation of oscillatory Sommerfeld integrals for electromagnetic and seismic problems in layered media^[7][8][9] and numerical solution to steady incompressible flow problems in fluid mechanics,^[10] as well as various different problems in neutron scattering,^[11] quantum mechanics^[12] and metallurgy.^[13]

• Tanh-sinh quadrature