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Given an algorithm producing the sequence of iterates $\{x_n\}$ converging to some value $x$, it is customary to express the manner in which the sequence converges using the concept of the order of convergence. Theoretically, this definition is useful, but in practice, one may have a complicated problem with a new method for which one may not be able to derive an expression for error term $x_{n+1}-x$ in terms of $x_n-x$. In this case, I would assume one computes the ratio involving the sequence of iterates $R_{n+1} = (x_{n+1} - x)/(x{_n-x})^a$, until one finds a value of a which causes the ratio to be about constant, in order to begin to get some insight into what the nature of the convergence is.

My question is whether one calls the value of $a$ obtained in this manner the order of convergence?

It seems like since a convergent sequence is Cauchy this method should be legitimate and the exponent obtained by this method should be the same (or very close) to the value one would obtained if the true value of $x$ were known. However, I am not absolutely sure this is true.

Thanks, Matt

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For comparison, Newton's algorithm typically has $x_{n+1} - x \approx (x_n - x)^2$, so your $a$ would be uninteresting. –  Hurkyl Sep 1 '12 at 20:28
    
That was an editing error: it is only the denominator which is raised to the power a –  Matt Brenneman Sep 1 '12 at 23:42
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