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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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    $\begingroup$ @johhn: Sounds less efficient than the Newton Method, $1.618$ versus $2$. But typically each iteration is faster, since we do not have to evaluate a derivative. $\endgroup$ Commented Apr 27, 2012 at 21:50
  • $\begingroup$ The advantage of Broyden (secant) methods is even more pronounced in the solution of equations in many variables, since derivatives (Jacobians) are even more complicated in that setting, and secant methods sidestep that problem nicely. $\endgroup$ Commented Apr 28, 2012 at 2:48
  • $\begingroup$ @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. $\endgroup$ Commented Apr 28, 2012 at 14:55
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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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    $\begingroup$ I guess the OP is talking about optimization and so about finding zeros of $f'$. $\endgroup$
    – lhf
    Commented Apr 28, 2012 at 0:31

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