Secant method

Remarks

The error formula comes from writing $x_{k+1} - r$ with divided differences: $e_{k+1} = e_k e_{k-1} f[x_{k-1}, x_k, r] / f[x_{k-1}, x_k]$, and the mean value theorem for divided differences turns these into $f''(\xi_k)/2$ and $f'(\eta_k)$. Let $M$ be the maximum of $\abs{f''}$ over $J$ divided by twice the minimum of $\abs{f'}$ over $J$. Choose $\delta$ so that $f' \ne 0$ on $J$ and $M \delta < 1$; then $\abs{e_{k+1}} \le M \delta \abs{e_k}$, which gives clause 1. Example: for $f(x) = x^2 - 2$, $x_0 = 1$, $x_1 = 2$, the iterates are $4/3$, $1.4$, $1.41463$, $1.41421$, approaching $\sqrt{2}$. The condition $f'(r) \ne 0$ matters. At a double zero such as that of $f(x) = x^2$, the method still converges from nearby points on the same side of $0$, but only linearly, with ratio tending to $(\sqrt{5} - 1)/2$. Far from a zero the method can fail, for example when $f(x_k) = f(x_{k-1})$ and the step is undefined.

Going deeper

For a student

The error recursion $e_{k+1} \approx C e_k e_{k-1}$ with $C = f''(r)/(2f'(r))$ is the source of the superlinear convergence of the secant method, with order the golden ratio $(1 + \sqrt{5})/2 \approx 1.618$. Newton's method has order $2$ but needs $f'$ at each step; per function evaluation, the secant method is often faster. Keeping a bracket instead of the last two points gives regula falsi, which is safe but can be slow.