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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

\[\operatorname{agm}(a,b) = \frac{(a+b) \pi}{4 K(z^2)},\]

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

\[\frac{\pi}{4 K(z^2)} = \tfrac{1}{2} - \tfrac{1}{8} z^2 - \tfrac{5}{128} z^4 - \tfrac{11}{512} z^6 - \tfrac{469}{32768} z^8 + \ldots\]

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

\[ \begin{align}\begin{aligned}\left|\frac{M(z+h) - M(z)}{h} - M'(z)\right| \le \frac{C D^2 h}{1 - D h}\\\left|\frac{M(z+h) - M(z-h)}{2h} - M'(z)\right| \le \frac{C D^3 h^2}{1 - D h}\\\left|\frac{M(z+h) + M(z-h)}{2} - M(z)\right| \le \frac{C D^2 h^2}{1 - D h}\end{aligned}\end{align} \]

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

\[ \begin{align}\begin{aligned}|M'(z+\varepsilon)| \le \frac{\max(1, |z|+|\varepsilon|+r)}{r}\\|M''(z+\varepsilon)| \le \frac{2 \max(1, |z|+|\varepsilon|+r)}{r^2}\end{aligned}\end{align} \]

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

\[-2 z (z^2-1) c_2 = (3z^2-1) c_1 + z c_0,\]
\[-(k+2)(k+3) z (z^2-1) c_{k+3} = (k+2)^2 (3z^2-1) c_{k+2} + (3k(k+3)+7)z c_{k+1} + (k+1)^2 c_{k}\]

in general, and

\[-(k+2)^2 c_{k+2} = (3k(k+3)+7) c_{k+1} + (k+1)^2 c_{k}\]

when \(z = 1\).