The Secant Method is basically a way of replacing the second derivative of a function with an approximation. So $$f''(x) \approx \frac{f'(x_2)-f'(x_1)}{x_{2}-x_{1}}$$
We then use this in Newton's formula $$x_{k+1} = x_{k}-\frac{f'(x_k)}{f''(x_k)}$$
It turns out that the secant method converges with rate equal to the golden ratio. Is this just a coincidence?
Best Answer
It is not a coincidence. It comes from the similarity from one stage of the root finding to the next and the fact that $\frac 1{\phi}=\phi-1$. A good discussion is in section 9.2 of Numerical Recipes