Definition:Arithmetic-Geometric Mean

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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:

\(\ds \forall k \in \N: \, \) \(\ds \) \(\) \(\ds \)
\(\ds a_{k + 1}\) \(=\) \(\ds \dfrac {a_k + b_k} 2\)
\(\ds b_{k + 1}\) \(=\) \(\ds \sqrt {a_k b_k}\)


\(\ds a_0\) \(=\) \(\ds a\)
\(\ds b_0\) \(=\) \(\ds b\)

The above process is known as the arithmetic-geometric mean iteration.

Also see

  • Results about the arithmetic-geometric mean can be found here.