bisection.m

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bisection.m#

This is a bisection method, specifically designed to determine when rays intersect a contour such that \(\omega|g(z)-g(\xi)|=C\), where \(g\) is the phase function, \(\omega\) is the frequency parameter and \(\xi\) is a stationary point.

The function to minimise is therefore

\[ f(r) = \omega|g(\xi + r\mathrm{e}^{\mathrm{i}\theta})-g(\xi)| - C, \]

and we’re seeking \(f(r)=0\) for some \(r\in[a,b]\) (here \(a\) and \(b\) are not the original integration endpoints.)

This function will only be called when the main rootfinding algorithm (which uses a companion matrix approach) fails.

[r, numIterations] = bisection(f, a, b, tol)

Inputs#

  • f is the function \(f\) defined above

  • a and b correspond to the search range \([a,b]\)

  • tol is the parameter which determines when the bisection method will halt, specifically when \(|f(c)|<\)tol.

Outputs#

  • c is the zero of \(f\), as defined above.

  • numIterations is the number of iterations that the bisection method ran for.

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