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.
Define . By the arithmetic-geometric inequality, we
have . By the defining recursions, we have
Since , we have , and , hence
From this inequality
we may conclude that as , which , by the definition of ,
is equivalent to
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 |
---|---|
Canonical name | ConvergenceOfArithmeticgeometricMean |
Date of creation | 2013-03-22 17:09:46 |
Last modified on | 2013-03-22 17:09:46 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 13 |
Author | rspuzio (6075) |
Entry type | Theorem![]() |
Classification | msc 33E05 |
Classification | msc 26E60 |