convergence of arithmetic-geometric mean
In this entry, we show that the arithmetic-geometric mean converges. By the arithmetic-geometric means inequality, we know that the sequences of arithmetic and geometric means are both monotonic and bounded, so they converge individually. What still needs to be shown is that they converge to the same limit.
Since , we have , and , hence
From this inequality
Not only have we proven that the arithmetic-geometric mean converges, but we can infer a rate of convergence from our proof. Namely, we have that . Hence, we see that the rate of convergence of and to the answer goes as .
By more carefully bounding the recursion for above, we may obtain better estimates of the rate of convergence. We will now derive an inequality. Suppose that .
Set (so we have ).
Thus, because , we have
From this equation, we may derive the bound
This is a much better bound! It approaches zero far more rapidly than any exponential function, so we have superlinear convergence.
|Title||convergence of arithmetic-geometric mean|
|Date of creation||2013-03-22 17:09:46|
|Last modified on||2013-03-22 17:09:46|
|Last modified by||rspuzio (6075)|