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

rate (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:

enter image description here (2)

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

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