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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?

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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

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    @john: I was remembering the minimization problem, where there is a good discussion in section 10.1. For root finding, there is just a claim of $\phi$ but no proof.2012-04-28
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The secant method replaces the first derivative, not the second.

Also, the Newton's method iteration is $x_{n+1} = x_n - f(x_n)/f'(x_n)$.

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    I guess the OP is talking about optimization and so about finding zeros of $f'$.2012-04-28