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. 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]
