Sufficient conditions for secant method to converge

Question

Are there some sufficient conditions for secant method $$x_{k+1}=x_k-\dfrac{x_k-x_{k-1}}{f(x_k)-f(x_{k-1})}f(x_k)$$

to converge?

Answer 1

You can easily deduce a local result. The error satisfies $$ e_{n+1} = -\dfrac{f''(\xi_n)}{2f'(\eta_n)} e_{n} e_{n-1}. $$

If $f\in C^2$ and $f'(z)\ne 0$, the method will converge if $x_0, x_1$ are sufficiently close to $z$. the rate of convergence will be $\alpha = (1+\sqrt{5})/2 \approx 1.618$.