implement the agm(x,y) function

[rwst]

https://en.wikipedia.org/wiki/Arithmetic-geometric_mean

Pari has a numeric implementation:

? 1/agm(1,sqrt(2))
%1 = 0.83462684167407318628142973279904680900

but Wikipedia provides a closed form integral expression, and if we had the "complete elliptic integral of the first kind" this would be even simpler.

Numerically there is sage.rings.real_mpfr.RealNumber.

CC: @mforets

Component: calculus

Issue created by migration from https://trac.sagemath.org/ticket/16202

[kcrisman]

comment:1

Do you mean elliptic_kc? This is indeed in Maxima, though not yet a "Sage symbolic function". See also the symbolics page on Trac where a few things about this are mentioned.

Oh, I see what you mean about the elliptic - like this Rosetta stone. Anyway, I would think that we can do this fairly easily - also note mpmath has the agm and the elliptic integral in question, and mpmath is probably a go-to for numerical evaluation of our most recent implementations of special functions.

[kcrisman]

comment:2

Sorry, to clarify - if we implement elliptic_kc as a symbolic function, you could do this easily as you say, or we can try to combine this with mpmath as well.

[rwst]

Description changed:

--- 
+++ 
@@ -7,3 +7,5 @@
 %1 = 0.83462684167407318628142973279904680900
 ```
 but Wikipedia provides a closed form integral expression, and if we had the "complete elliptic integral of the first kind" this would be even simpler.
+
+Numerically there is `sage.rings.real_mpfr.RealNumber`.

[fchapoton]

comment:7

There is already

sage: a=CDF(1)
sage: b=CDF(sqrt(2))
sage: 1/a.agm(b)
0.834626841674073

[rwst]

comment:8

Also

            sage: RBF(sqrt(2)).agm(1)^(-1)
            [0.83462684167407 +/- 3.9...e-15]