Definition:Arithmetic-Geometric Mean/Iteration
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Definition
Let $a$ and $b$ be numbers.
Let $\sequence {a_n}$ and $\sequence {b_n}$ be defined as the recursive sequences:
\(\ds \forall k \in \N: \, \) | \(\ds \) | \(\) | \(\ds \) | |||||||||||
\(\ds a_{k + 1}\) | \(=\) | \(\ds \dfrac {a_k + b_k} 2\) | ||||||||||||
\(\ds b_{k + 1}\) | \(=\) | \(\ds \sqrt {a_k b_k}\) |
where:
\(\ds a_0\) | \(=\) | \(\ds a\) | ||||||||||||
\(\ds b_0\) | \(=\) | \(\ds b\) |
The above process is known as the arithmetic-geometric mean iteration.
Also see
- Results about the arithmetic-geometric mean can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arithmetic-geometric mean iteration
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): mean: 4.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): mean: 4.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): arithmetic-geometric mean iteration
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): arithmetic-geometric mean iteration