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. 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]

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