Convergence

Since at the end of each iteration we have an interval known to contain the solution, and since we choose the middle of this interval ($\varphi$-wise), we can use its half-size as a convergence criterion. We stop the iterations when:

$$\frac{\vert\varphi_{n}-\tilde{\varphi}_n\vert}{2}<\varepsilon_c$$ (14)

where $\varepsilon_c$ is the convergence threshold.

When this threshold is met, we use $\varphi=\varphi_{n+1}$ and we compute $h$ using the following expression which is true when $\varphi$ has been found:

$$h = r\cos\varphi+z\sin\varphi-a_e\sqrt{1 - f(2-f)\sin^2\varphi}$$ (15)