fsolve or nSolve with the option secant_solver

The secant methode is a simplified version of the Newton method. The computation of x₁ is done using the Newton method. The computation of f'(x_n), n > 1 is done approximatively. This method is used when the computation of the derivative is expensive:

x_i+1 = x_i - $${\frac{{ f(x_i)}}{{f'_{est}}}}$$,    f'_est = $${\frac{{f(x_i) - f(x_{i-1})}}{{(x_i - x_{i-1})}}}$$

The convergence for roots of multiplicity 1 is of order (1 + $\sqrt{5}$)/2 $\approx$ 1.62....
Input :

fsolve((cos(x))=x,x,-1..1,secant_solver)

Output :

[0.739085078239,0.739085137844]

Input :

fsolve((cos(x))=x,x,0,secant_solver)

Output :

0.739085133215