getSmallestSupsetBall

As part of the algorithm, we determine non-oscillatory balls. These are chosen roughly such that

\[ \omega|g(z)-g(\xi)| \leq C_{\mathrm{ball}} \]

where \(\xi\) is the centre of the ball, and \(z\) is any point inside the ball. Clearly the region \(\Omega\) where this holds is not a ball - we choose either the smallest ball containing \(\Omega\), or the largest ball inside \(\Omega\), determined by isLargestBall.

This subroutine finds a single non-oscillatory ball. Specifically, given the centre of the ball, and information about the phase and frequency, the algorithm determines the radius.

r = getSmallestSupsetBall(phaseCoeffs, freq, stationaryPoint, ...
                                    numOscs, numRays, isLargestBall, ...
                                    imag_thresh)

Inputs

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

• freq: Frequency parameter \(\omega\)

• stationaryPoint: The stationary point \(\xi\) at the centre of the ball

• numOscs: An approximate number of oscillations permitted inside of the ball, denoted \(C_{\mathrm{ball}}\)

• numRays: The number of angles at which the rootfinding algorithm getRGivenTheta is performed.

• isLargestBall: Boolean parameter, if true, we choose the largest ball at which the cost function is minimised.

• imag_thresh: A threshold for which roots are considered to be imaginary.

Outputs

r is the radius of the ball.