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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)?

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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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