ContourSD.m

ContourSD.m#

A Matlab class designed to represent steepest descent contours. Recall that a steepest descent contour \(h_\eta(p)\) starts at the point \(h_\eta(0)=\eta\), and follows the path such that \(g(h_\eta(p))=\mathrm{i}p\). Along such a path, the integrand of our original integral is exponentially decaying as \(p\) increases.

These contours can be finite or infinite, but in practice they must be approximated for finite values of \(p\).

Properties#

  • startPoint: Start of approximate contour, i.e. an exit. This is denoted \(\eta\) in [Gibbs et al., 2024].

  • endPoint: End of approximate contour.

  • startCoverIndex: Index of the ball in which contour starts.

  • endCoverIndex: (Possible) index of ball where contour ends. Can be blank.

  • coarsePath: Array of complex points approximating contour \(h_\eta\).

  • coarseParam: Parameter \(p>0\) in \(h_\eta(p)\), corresponding to above array.

  • length: max value of the parameter \(p\), can be infinite for infinite paths.

  • endValley: (Possible) valley where path converges. Can be blank.

  • phaseCoeffs: Coefficients of polynomial phase function \(g\).

  • phaseDerivs: Coefficients of derivatives of polynomial phase function \(g'\).

  • ICs: initial conditions for ODE solve, \((h_\eta(0), h_\eta'(0))\)

Methods#

plot#

This plots the approximate contour.

getQuad#

This produces a quadrature rule along a given contour.

[hArray, wArray, newtonSuccess] = getQuad(self, freq, numPts, QuadParams)

Inputs#

  • freq: The frequency parameter \(\omega\)

  • numPts: The number of desired quadrataure points along the contour, \(N\).

  • QuadParams: A structure containing parameters for the quadrature rule; the approximation of the quadrature points, the type of quadrature, etc.

Outputs#

  • hArray: \(N\) quadrature points

  • wArray: \(N\) quadrature weights

  • newtonSuccess: a boolean array of length \(N\), each entry is true if the Newton iteration converged for the corresponding quadrature point.

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