On Mon, Aug 10, 2009 at 5:59 AM, John Cremona <[email protected]>wrote:
>
> Surely one option would be to implement elliptic_e ourselves in Sage;
> or use the pari library, if it is there? Some elliptic functions are
> there. If elliptic_e is one of the ones which can be computed
> efficiently usn the AGM it should not be hard.
>
Fredrick posted a 1-line implementation using mpmath, which I'll copy here
(see below). Our time might be better spent changing elliptic_e to use this
instead of maxima for numerical evaluation, if possible. Note that there
still might be subtle reasons that wouldn't 100% work though, e.g., what if
Maxima numerically evaluates elliptic_e internally in the course of doing
some symbolic solve, integral or simplification? If so, by implementing a
numerical elliptic_e ourselves, but still depending on maxima for some
operations, we've simply hidden a potential bug. This is one reason we need
to fully implement symbolic solving, integrals, etc. ourselves (and/or using
Sympy).
"mpmath doesn't have incomplete elliptic integrals yet, but they can be
computed using the Appell F1 function or directly using numerical
quadrature. (The following implementations assume |re(z)| < pi/2 and
possibly other restrictions on the variables.)
>>> from mpmath import mp, mpf, sin, appellf1, quad
>>>
>>> def E(z,m):
... return sin(z)*appellf1(0.5,0.5,-0.5,1.5, sin(z)**2, m*sin(z)**2)
...
>>> def E2(z,m):
... return quad(lambda t: (1-m*sin(t)**2)**0.5, [0,z])
...
>>> mp.dps = 50; mp.pretty = True
>>> E(0.5, mpf('0.1'))
0.49801139449883153311546104061744810584963105068054
>>> E2(0.5, mpf('0.1'))
0.49801139449883153311546104061744810584963105068054
Which is the same as Mathematica's output, so this looks like a Maxima bug.
Fredrik"
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