Secant method

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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 meanbutton.png

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.


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