# Algorithms for the arithmetic-geometric mean¶

With complex variables, it is convenient to work with the univariate function \(M(z) = \operatorname{agm}(1,z)\). The general case is given by \(\operatorname{agm}(a,b) = a M(1,b/a)\).

## Functional equation¶

If the real part of *z* initially is not completely nonnegative, we
apply the functional equation \(M(z) = (z+1) M(u) / 2\)
where \(u = \sqrt{z} / (z+1)\).

Note that *u* has nonnegative real part, absent rounding error.
It is not a problem for correctness if rounding makes the interval
contain negative points, as this just inflates the final result.

For the derivative, the functional equation becomes \(M'(z) = [M(u) - (z-1) M'(u) / ((1+z) \sqrt{z})] / 2\).

## AGM iteration¶

Once *z* is in the right half plane, we can apply the AGM iteration
(\(2a_{n+1} = a_n + b_n, b_{n+1}^2 = a_n b_n\)) directly.
The correct square root is given by \(\sqrt{a} \sqrt{b}\),
which is computed as \(\sqrt{ab}, i \sqrt{-ab}, -i \sqrt{-ab}, \sqrt{a} \sqrt{b}\)
respectively if both *a* and *b* have positive real part, nonnegative
imaginary part, nonpositive imaginary part, or otherwise.

The iteration should be terminated when \(a_n\) and \(b_n\) are close enough.
For positive real variables, we can simply take lower and upper bounds
to get a correct enclosure at this point. For complex variables, it is shown
in [Dup2006], p. 87 that, for *z* with nonnegative real part,
\(|M(z) - a_n| \le |a_n - b_n|\), giving a convenient error bound.

Rather than running the AGM iteration until \(a_n\) and \(b_n\) agree to \(p\) bits, it is slightly more efficient to iterate until they agree to about \(p/10\) bits and finish with a series expansion. With \(z = (a-b)/(a+b)\), we have

valid at least when \(|z| < 1\) and \(a, b\) have nonnegative real part, and

where the tail is bounded by \(\sum_{k=10}^{\infty} |z|^k/64\).

## First derivative¶

Assuming that *z* is exact and that \(|\arg(z)| \le 3 \pi / 4\),
we compute \((M(z), M'(z))\) simultaneously using a finite difference.

The basic inequality we need is \(|M(z)| \le \max(1, |z|)\), which is an immediate consequence of the AGM iteration.

By Cauchy’s integral formula, \(|M^{(k)}(z) / k!| \le C D^k\) where
\(C = \max(1, |z| + r)\) and \(D = 1/r\), for any \(0 < r < |z|\) (we
choose *r* to be of the order \(|z| / 4\)). Taylor expansion now gives

assuming that *h* is chosen so that it satisfies \(h D < 1\).

The forward finite difference would require two function evaluations at doubled precision. We use the central difference as it only requires 1.5 times the precision.

When *z* is not exact, we evaluate at the midpoint as above
and bound the propagated error using derivatives.
Again by Cauchy’s integral formula, we have

assuming that the circle centered on *z* with radius \(|\varepsilon| + r\)
does not cross the negative half axis. We choose *r* of order \(|z| / 2\)
and verify that all assumptions hold.

## Higher derivatives¶

The function \(W(z) = 1 / M(z)\) is D-finite. The coefficients of \(W(z+x) = \sum_{k=0}^{\infty} c_k x^k\) satisfy

in general, and

when \(z = 1\).