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Assuming $p$ is the exact solution and $p_n$ is a numerical approximation. My question is that why most of the numerical analysis books using $$ \frac{|p_n-p_{n-1}|}{|p_n|} $$ to approximate the following relative error $$ \frac{|p_n-p|}{|p|} $$ ?

Dante
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    Do they really claim it approximates the relative error? Or do they just say it can be used for a stopping criterion. If the first quantity is small, then the iterates aren't changing much anymore, suggesting the algorithm has converged. – littleO Sep 25 '13 at 23:03
  • @littleO I think you could go ahead and post your comment as an answer – davidlowryduda Oct 24 '13 at 02:43

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The quantity $\frac{|p_n - p_{n-1}|}{|p_n|}$ does not necessarily approximate the relative error $\frac{|p_n - p|}{|p|}$ (and I don't think numerical analysis books claim that it does). However, $\frac{|p_n - p_{n-1}|}{|p_n|}$ can be used in a stopping criterion. If this quantity is small, then the iterates aren't changing much anymore, which suggests that the algorithm has converged.

littleO
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