# Secant method

A method to find a root of a univariate function, say F. The iterate is

$LaTeX: x^{k+1} = x^k - \frac{\mbox{F}(x^k) [x^k - x^(k-1)]}{\mbox{F}(x^k) - \mbox{F}(x^{k-1})}.$

If $LaTeX: \textstyle \mbox{F} \in \mbox{C}^2 \mbox{ and F}''(x) \ne 0,$ the order of convergence is the golden mean, say g (approx.= 1.618), and the limiting ratio is:

$LaTeX: \left | \frac{2 \mbox{F}'(x)}{\mbox{F}''(x)} \right |^{g-1}$

# Golden mean

The golden mean, or golden ratio is the positive solution to the quadratic equation, $LaTeX: x^2 + x - 1 = 0$, which is $LaTeX: (\sqrt{5}-1)/2$, or approximately $LaTeX: 0.618$. This has the proportionality property: $LaTeX: x : 1 = (1-x) : x$, which defines the placements of evaluations for the golden section search.