getExteriorBalls.m

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getExteriorBalls.m#

This function performs many important topological operations at an early stage of the PathFinder algorithm, by calling more specialised subroutines. For a full mathematical description, see [Gibbs et al., 2024].

Here is a high level overview:

  • Stationary points of the phase function \(g\) are determined.

  • These stationary points are merged, if they are sufficiently close.

  • Balls are placed around each of these stationary points, such that the interior of the ball may be considered non-oscillatory.

  • Upon creation of each ball, exits are placed around the circumference. Any exit inside another ball is immediately deleted.

[covers, endPointIndices]...
                = getExteriorBalls(phaseFn, freq, stationaryPointArray, ...
                    isInfEndptArray, a, b, Cosc, phaseCoeffs, ...
                    ballMergeTol, numRays, isInteriorBalls, imagThresh, ...
                    useMex)

Inputs#

  • phaseFn: A function handle for the polynomial phase \(g(z)\).

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

  • stationaryPointArray: Stationary points of \(g\), i.e. \(\xi\) where \(g'(\xi)=0\).

  • isInfEndptArray: A boolean array of length two, describing if the endpoints \(a\) and \(b\) are infinite

  • a: The endpoint \(a\). If finite, this is the position in \(\mathbb{C}\); if infinite, this is the complex angle \(\theta\) such that \(a=\exp(\mathrm{i}\theta)\).

  • b: The endpoint \(b\), defined analogously to \(a\) above.

  • Cosc: The parameter corresponding to the number of oscillations.

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

  • ballMergeTol: Parameter that decides how close stationary points must be to be merged.

  • numRays: The number of angles to divide each ball when determining an appropriate radius. See getSmallestSupsetBall.m for more information.

  • isInteriorBalls: Also a parameter used by getSmallestSupsetBall.m.

  • imagThresh: Size of permitted imaginary component in polynomial rootfinding process, see getRGivenTheta.m.

  • useMex: Boolean, true if MEX version of rootfinding code is to be used.

Outputs#

  • covers is a cell array of objects of Ball type.

  • endPointIndices are the indices of the cell array corresponding to balls centred at finite endpoints. For example, if \(\{a,b\}\) are finite, endPointIndices is \([1,2]\).

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