Secant method
The secant method uses secant lines of the graph of a locally linear function to approximate its real or complex roots. For a function 
, the approximations are defined recursively by ![\[x_{i+1} = \frac{x_{i-1}f(x_i) - x_if(x_{i-1})}{f(x_i) - f(x_{i-1})}.\]](http://latex.artofproblemsolving.com/2/5/e/25eb5ceec2da8fc0f66ccfeb7d1c21c9404cd153.png)
The values 
and 
used initially in the recursion are guesses.
Derivation
Since the secant line between points 
and 
is a linear function, it has exactly one root (unless 
, in which case the method fails). The root of the secant line serves as 
, an improved approximation to the root of .
The slope of the secant line is ![\[\frac{f(x_i) - f(x_{i-1})}{x_i - x_{i-1}}.\]](http://latex.artofproblemsolving.com/a/4/7/a4700982baddf48cfed9c443ef63bcdb5343a5dc.png)
The point 
is on the secant line, so ![\[\frac{f(x_i) - f(x_{i-1})}{x_i - x_{i-1}} = \frac{0 - f(x_i)}{x_{i+1} - x_i}.\]](http://latex.artofproblemsolving.com/0/1/b/01b0b2ca5f2692b59442aea4b20daa0ec002ed23.png)
Rearranging, ![\[x_{i+1} = x_i - \frac{f(x_i)(x_i - x_{i-1})}{f(x_i) - f(x_{i-1})}.\]](http://latex.artofproblemsolving.com/4/2/4/424c3f48c45f51dbe0dc1e1482a9ef5273577b20.png)
The reformulation ![\[x_{i+1} = \frac{x_{i-1}f(x_i) - x_if(x_{i-1})}{f(x_i) - f(x_{i-1})}\]](http://latex.artofproblemsolving.com/9/4/0/940182bdc61c5544916f7cf112bd52a12071d67e.png)
more clearly shows the symmetry between 
and 
in the calculation of .
Failure cases
The secant method fails when there are two adjacent estimates 
and for which 
(analogous to reaching a zero derivative in Newton's method), and like Newton's method may converge to an unexpected root, cycle periodically, or diverge.
