The secant method is a root-finding algorithm that 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.
The secant method is implemented in the Wolfram
Language as the undocumented option Method -> Secant in FindRoot[eqn,
x, x0, x1
].
When the algorithm does converge, its order of convergence is
 |
(1)
|
where 
is a constant and 
is the golden ratio.
 |
(2)
|
 |
(3)
|
 |
(4)
|
so
 |
(5)
|
The secant method can be implemented in the Wolfram Language as
SecantMethodList[f_, {x_, x0_, x1_}, n_] :=
NestList[Last[#] - {0, (Function[x, f][Last[#]]*
Subtract @@ #)/Subtract @@
Function[x, f] /@ #}&, {x0, x1}, n]
