Secant Method

A Root-finding algorithm which assumes a function to be approximately linear in the region of interest. Each improvement is taken as the point where the approximating line crosses the axis. The secant method retains only the most recent estimate, so the root does not necessarily remain bracketed. When the Algorithm does converge, its order of convergence is

$$\lim_{k\to\infty} \vert\epsilon_{k+1}\vert \approx C\vert\epsilon\vert^\phi,$$ (1)

where $C$ is a constant and $\phi$ is the Golden Mean.

$$f'(x_{n-1}) \approx {f(x_{n-1})-f(x_{n-2})\over x_{n-1}-x_{n-2}}$$ (2)

$$f(x_n)\approx f(x_{n-1}) + f'(x_n)(x_n-x_{n-1}) =0$$ (3)

$$f(x_{n-1})+{f(x_{n-1})-f(x_{n-2})\over x_{n-1}-x_{n-2}}(x_n-x_{n-1})=0,$$ (4)

so

$$x_n=x_{n-1}-{f(x_{n-1})(x_{n-1}-x_{n-2})\over f(x_{n-1})-f(x_{n-2})}.$$ (5)

See also False Position Method

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Secant Method, False Position Method, and Ridders' Method.'' §9.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 347-352, 1992.