Let $g(x)=0.5(x+a/x)$. Determine the order of convergence and the asymptotic error constant of the sequence $p_n=g(p_{n-1})$ toward $x=a^{.5}$.

This is a problem in our homework in the class Numerical Analysis and our teacher never went over anything like this at all so I have no idea. You don't have to give me the answer, but I could really use to advice on how to approach this problem.


Here's hint:

$$\left|g(x)-\sqrt{a}\right| = \frac{\left(x-\sqrt{a}\right)^2}{2 x}.$$

Thus, if $|x-\sqrt{a}|$ is small, then $|g(x)-\sqrt{a}|$ is small squared - i.e. really small.

Incidentally, this sequence arises from the application of Newton's method to the function $f(x)=x^2-a$.

  • $\begingroup$ what about the asymptotic error? $\endgroup$ – Brian Sep 4 '14 at 3:35
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    $\begingroup$ @Brian Think about it. The hint addresses both the fact of convergence and the rate of convergence. At some point you've got to stop blaming your teacher with copy-pasted statements and take some responsibility for your learning. $\endgroup$ – user147263 Sep 4 '14 at 4:08

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