fsolve or nSolve with the option steffenson_solver

The Steffenson method is generically the fastest method.
It combines the Newton method with a "delta-two" Aitken acceleration : with the Newton method, we obtain the sequence x_i and the convergence acceleration gives the Steffenson sequence

R_i = x_i - $${\frac{{(x_{i+1} - x_i)^2}}{{ (x_{i+2} - 2 x_{i+1} + x_{i})}}}$$

Input :

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

Output :

0.739085133215