checkEndpointWidth.m

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checkEndpointWidth.m#

Determines if an integral over a bounded contour can be considered non-oscillatory.

As described in the paper [Gibbs et al., 2024], this is a preliminary step to determine if the original integral can be considered non-oscillatory. To achieve this, we place temporary balls around both endpoints \(\{a,b\}\), based on the same criteria used to determine radii throughout the algorithm. If the balls overlap, this is an indication that there are only a few oscillations between the endpoints.

isOverlapping = checkEndpointWidth(a, b, phaseCoeffs, freq, numOscs, numRays, isInteriorBal, imagThresh, useMex)

Inputs#

  • a and b are the endpoints of the original integral

  • phaseCoeffs is a vector of coefficients of the polynomial phase function in the original integral

  • freq is the frequency parameter of the original integral

  • numOscs is the parameter which governs the size of the balls; the larger this value, the more oscillations are permitted inside of the balls.

  • numRays the number of rays fired from the endpoints \(\{a,b\}\) when determining the non-oscillatory ball around them.

  • isInteriorBall is a boolean flag which determines if the non-oscillatory ball contains the non-oscillatory region, or is contained within it. Practically, it corresponds to choosing the shortest ray (interior) or the longest ray (exterior) which intercepts the boundary of the non-oscillatory region in the bisection method.

Outputs#

  • isOverlapping is a boolean flag, true if the non-oscillatory balls around the endpoints \(a\) and \(b\) overlap.

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