The iteration function C(mi) depends on the constants phi and sigma. The standard sufficient condition for local convergence of the fixed point method is that |C'(mi)| < 1 in a neighborhood of the fixed point, which ensures what is usually called linear convergence. So the first step is to find values of phi and sigma such that this condition is true. I assume (but this is an interpretation, since this terminology is less standard) that by quadratic convergence it is meant that C'=0 at the fixed point and by cubic convergence that C'' =0 at the fixed point.

So this faster convergence can be achieved if one can find phi, sigma values such that these further sufficient conditions are true.

Ohh.

It did not occur to me that C” should be zero as well.