PathFinder Documentation

PathFinder is a Matlab/Octave toolbox for the numerical evaluation of highly oscillatory integrals. Specifically, PathFinder can efficiently evaluate integrals of the form

\[ I = \int_{a}^b f(z)\exp(\mathrm{i}\omega g(z)) \mathrm{d}z \]

where \(g\) is a polynomial, \(f\) is entire (analytic everywhere in \(\mathbb{C}\)), \(\omega>0\) is a frequency parameter, and the endpoints \(a\) and \(b\) may be finite or infinite. Further, it is assumed that \(I\) is a convergent integral and that \(|f(z)|\) grows sub-exponentially as \(|z|\to\infty\).

PathFinder is based on steepest descent contour deformation, but it can be easily used without a deep understanding of the underlying mathematics; it is sufficient to understand the conditions in the previous paragraph. This document intends to provide a full explanation of how to use PathFinder. For a full explanation of the underlying mathematics, the interested reader is referred to [Gibbs et al., 2024].