Timeline for nth-order generalizations of the arithmetic-geometric mean
Current License: CC BY-SA 3.0
14 events
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| Nov 27, 2011 at 17:26 | history | edited | KConrad | CC BY-SA 3.0 |
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| Nov 27, 2011 at 17:19 | history | edited | KConrad | CC BY-SA 3.0 |
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| Nov 27, 2011 at 17:13 | history | edited | KConrad | CC BY-SA 3.0 |
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| Nov 27, 2011 at 0:33 | history | edited | KConrad | CC BY-SA 3.0 |
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| Nov 15, 2011 at 1:57 | history | edited | KConrad | CC BY-SA 3.0 |
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| Nov 15, 2011 at 1:46 | history | edited | KConrad | CC BY-SA 3.0 |
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| Nov 15, 2011 at 1:36 | history | edited | KConrad | CC BY-SA 3.0 |
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| Nov 14, 2011 at 21:07 | comment | added | Will Sawin | If you want an estimate of the convergence speed, the following argument can probably be made rigorous: Let $a_i=a+\epsilon\delta_i$ for small $\epsilon$.We can write the arithmetic mean as $a+\epsilon \sum_i \delta_i/n$ and all the other means as $a+\epsilon \sum_i \delta_i/n+O(\epsilon^2)$, using their Taylor series ,so one can write the second iteration in terms of the first with $\epsilon'=O(\epsilon^2)$ | |
| Nov 14, 2011 at 18:04 | comment | added | Will Sawin | If you don't care about getting a good bound, finding a proof isn't very hard. Suppose $x$ and $y$ are the smallest and largest values of $a_i$, with $d=y−x$. We want to bound $A_1-\sqrt[n]{A_n}$ as a function of $d$, which we can do by just noticing that $A_1\leq y-d/n$ and $A_n\geq x^n$. Therefore, $d$ decreases at least exponentially, and they all have the same limit. | |
| Sep 5, 2010 at 2:06 | comment | added | KConrad | Unfortunately, as I wrote in my answer above, I can't remember where I saw the proof that the iteration converges. If you can find such a reference then I am sure the proof will show you why the convergence is fast, e.g., perhaps after some specific number of iterations the difference between the largest and smallest number should be cut down by at least a factor of 2. Consider posting a new question asking for a reference on the convergence. | |
| Sep 3, 2010 at 19:46 | comment | added | J. M. isn't a mathematician | Nice, this is my first time to hear about this generalization of the AM-GM inequality. Thanks KConrad! You wouldn't happen to know anything about convergence rate now, would you? (Though experimentation shows that it never takes more than six iterations for 20 digit accuracy for positive real inputs; I haven't done tests for complex inputs). | |
| Sep 3, 2010 at 16:26 | history | edited | KConrad | CC BY-SA 2.5 |
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| Sep 3, 2010 at 16:17 | history | edited | KConrad | CC BY-SA 2.5 |
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| Sep 3, 2010 at 16:05 | history | answered | KConrad | CC BY-SA 2.5 |