Algorithm

A Regularised Numerical Steepest Descent (RNSD) method involves a combination of contour deformations and quadrature rules to evaluate oscillatory integrals. This is automated by the following basic interface:

nsd(γ0::Vector, f::Function, G::AbstractPhase, ω)

First, the algorithm first determines a quasi-SD contour, which is the union of three contour types:

• Finite contours inside non-oscillatory regions,
• Infinite contours going into valleys, regions where $|e^{i\omega g(z)}| \to 0$ as $z$ tends to the valley,
• Finite SD contours that connect two non-oscillatory regions.

Then, it applies suitable quadrature rules on each of these contours, e.g. Gaussian quadrature, and returns an approximated value.

Basic use

Consider an integral representation of the Airy function, such as (9.5.4) in DLMF, where the integration curve is an infinite curve $(\infty e^{-i\pi/3}, \infty e^{i\pi/3})$. To specify that endpoints are infinite, use infcontour = [true, true]. To visualise the quasi-SD contour followed, use plot_sd = true.

using NumericalSteepestDescent
Phase = PolynomialPhase(-im*[0,+2,0,1/3]) # We are evaluating Ai(-2)
a, b = -π/3, π/3 # (∞exp(-im*π/3), ∞exp(im*π/3))
f(z) = 1.0
ω = 1.0 # Frequency parameter
ai, fig = nsd([a, b], f, Phase, ω; infcontour = [true, true],
    plot_sd = true)

The colorbar shows $-\Im[g(\eta)]$, so blue regions are valleys at infinity, and red regions are hills (integrand grows exponentially). The red lines represent finite contours, blue lines SD contours, and green lines finite SD contours (not shown). Bold lines correspond to the quasi-SD deformed contour where quadrature rules are applied.

Advanced use

There is also a number of adjustable parameters