Actually calculating the rate of convergence of iteratvie methods when exact solution is unknown

Question

When solving a system of nonlinear equations using iterative methods, the rate of convergence usually is defined by the following formula:

(1)

where x* is the exact solution. However usually we don't know x*, so actually we use the following formula to calculate the rate of convergence of the method:

(2)

My question is how to validate that the alapha in (2) equals to that in (1)?

Answer 1

What you have is not an exact equivalence. It uses the fact that $$ \| x_{k+2}-x_{k+1} \| \approx \| x^\ast-x_{k+1} \| \\ \| x_{k+1}-x_{k} \| \approx \| x^\ast-x_{k} \| $$ because the next iterate is always much closer to the solution $x^\ast$ than the previous one.