Category:Arithmetic-Geometric Mean
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This category contains results about the arithmetic-geometric mean.
Definitions specific to this category can be found in Definitions/Arithmetic-Geometric Mean.
The arithmetic-geometric mean of two numbers $a$ and $b$ is the limit of the sequences obtained by the arithmetic-geometric mean iteration.
This is denoted $\map M {a, b}$.
Arithmetic-Geometric Mean Iteration
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.
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