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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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