Fixed Point Method

If we can say 

and we supose

X--->f(x)=0

X--->x+g(x)=0

 then

X=g(x)

Fixed point iteration for a general function g(x) for the four cases of interest. Generalizations of the two cases of positive slope shown are shown on the left, and illustrate monotonic convergence and divergence. The cases where g(x) has negative slope are shown on the right, and illustrate oscillating convergence and divergence. The top pair of panels illustrate strong and weak attractors, while the bottom pair of panels illustrate strong and weak repellers.