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98 lines
3.4 KiB
ReStructuredText
98 lines
3.4 KiB
ReStructuredText
.. _algorithms_agm:
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Algorithms for the arithmetic-geometric mean
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===============================================================================
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With complex variables, it is convenient to work with the univariate
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function `M(z) = \operatorname{agm}(1,z)`. The general case is given by
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`\operatorname{agm}(a,b) = a M(1,b/a)`.
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Functional equation
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------------------------------------------------------------------------------
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If the real part of *z* initially is not completely nonnegative, we
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apply the functional equation `M(z) = (z+1) M(u) / 2`
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where `u = \sqrt{z} / (z+1)`.
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Note that *u* has nonnegative real part, absent rounding error.
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It is not a problem for correctness if rounding makes the interval
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contain negative points, as this just inflates the final result.
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For the derivative, the functional equation becomes
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`M'(z) = [M(u) - (z-1) M'(u) / ((1+z) \sqrt{z})] / 2`.
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AGM iteration
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------------------------------------------------------------------------------
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Once *z* is in the right half plane, we can apply the AGM iteration
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(`2a_{n+1} = a_n + b_n, b_{n+1}^2 = a_n b_n`) directly.
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The correct square root is given by `\sqrt{a} \sqrt{b}`,
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which is computed as `\sqrt{ab}, i \sqrt{-ab}, -i \sqrt{-ab}, \sqrt{a} \sqrt{b}`
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respectively if both *a* and *b* have positive real part, nonnegative
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imaginary part, nonpositive imaginary part, or otherwise.
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It is shown in [Dup2006]_, p. 87 that, for *z* with nonnegative real part,
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`|M(z) - a_n| \le |a_n - b_n|`.
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A small improvement would be to switch to a series
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expansion for the last one or two steps.
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First derivative
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------------------------------------------------------------------------------
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Assuming that *z* is exact and that `|\arg(z)| \le 3 \pi / 4`,
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we compute `(M(z), M'(z))` simultaneously using a finite difference.
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The basic inequality we need is `|M(z)| \le \max(1, |z|)`, which is
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an immediate consequence of the AGM iteration.
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By Cauchy's integral formula, `|M^{(k)}(z) / k!| \le C D^k` where
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`C = \max(1, |z| + r)` and `D = 1/r`, for any `0 < r < |z|` (we
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choose *r* to be of the order `|z| / 4`). Taylor expansion now gives
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.. math ::
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\left|\frac{M(z+h) - M(z)}{h} - M'(z)\right| \le \frac{C D^2 h}{1 - D h}
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assuming that *h* is chosen so that it satisfies `h D < 1`.
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The forward finite difference requires two function evaluations
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at doubled precision. It would be more efficient to use a central difference
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(this could be implemented in the future).
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When *z* is not exact, we evaluate at the midpoint as above
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and bound the propagated error using derivatives.
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Again by Cauchy's integral formula, we have
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.. math ::
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|M'(z+\varepsilon)| \le \frac{\max(1, |z|+|\varepsilon|+r)}{r}
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|M''(z+\varepsilon)| \le \frac{2 \max(1, |z|+|\varepsilon|+r)}{r^2}
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assuming that the circle centered on *z* with radius `|\varepsilon| + r`
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does not cross the negative half axis. We choose *r* of order `|z| / 2`
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and verify that all assumptions hold.
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Higher derivatives
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-------------------------------------------------------------------------------
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The function `W(z) = 1 / M(z)` is D-finite. The coefficients of
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`W(z+x) = \sum_{k=0}^{\infty} c_k x^k` satisfy
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.. math ::
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-2 z (z^2-1) c_2 = (3z^2-1) c_1 + z c_0,
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.. math ::
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-(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}
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in general, and
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.. math ::
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-(k+2)^2 c_{k+2} = (3k(k+3)+7) c_{k+1} + (k+1)^2 c_{k}
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when `z = 1`.
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