Re: LegendreP error (bug?) in Mathematica

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• Subject: [mg81232] Re: LegendreP error (bug?) in Mathematica
• From: Roman <rschmied at gmail.com>
• Date: Mon, 17 Sep 2007 03:30:47 -0400 (EDT)
• References: <fcdf41\$pjg\$1@smc.vnet.net><fcg4rn\$ro6\$1@smc.vnet.net>

```John,
If you only need the spherical harmonics, but not the associated
Legendre polynomials, then it turns out that the GNU Scietific Library
does very well:
http://www.gnu.org/software/gsl/manual/html_node/Associated-Legendre-Polynomials-and-Spherical-Harmonics.html
I've tried this with very large numbers L and M, and the results were
very accurate (unlike the Mathematica results).
You can write a small C wrapper program that you then call using
Roman.

On Sep 15, 10:24 am, Roman <rschm... at gmail.com> wrote:
> Bob,
> Of course it is a precision problem. The question is, why does
> Mathematica's numerical algorithm for associated Legendre polynomials
> give wrong results for large m? The inaccuracies are far beyond what
> you'd expect from the limited precision of the input arguments. To be
> more specific, look at my previous example and series-expand around
> x=0.8:
>
> In[1] := Series[LegendreP[200, 43, x], {x, 4/5, 1}] // Normal // N
> Out[1] = 2.92564*10^97 + 1.37446*10^100 (-0.8 + x)
>
> >From the ratio of the coefficients you see that, roughly speaking, if
>
> I insert x=0.8 with a relative error of 10^(-16), then I expect a
> result with a relative error of 4*10^(-14). In general, I expect the
> relative error to be around 400 times greater in the result than in
> the argument. But not so for Mathematica's numerical algorithm. If you
> increase the relative precision, as you say,
>
> In[2] := LegendreP[200, 43, 0.8`50]
> Out[2] =
> 2.925642467661265646732164377813044273198`24.332438293726835*^97
>
> you get a result which is ok but more than 25 orders of magnitude less
> precise than the argument, far worse than the factor of 400 expected
> from the slope of the Legendre polynomial! So if you start with a
> machine number with 16 significant figures (MachinePrecision), and
> lose 25 in the calculation, then naturally you end up with garbage.
> But, instead of telling you that you are getting garbage, the
> algorithm returns a number with exaggerated precision:
>
> In[3] := LegendreP[200, 43, 0.8] // Precision
> Out[3] = MachinePrecision
>
> Thus I agree with John's complaint that Mathematica's numerical
> algorithm is not good. At least you should be getting a warning of
> some sort if the results are so bad.
>
> Unfortunately, it's no better if you use the GNU Scientific Library
> through MathLink, since the GSL manual says: "The following functions
> compute the associated Legendre Polynomials P_l^m(x). Note that this
> function grows combinatorially with l and can overflow for l larger
> than about 150. There is no trouble for small m, but overflow occurs
> when m and l are both large. Rather than allow overflows, these
> functions refuse to calculate P_l^m(x) and return GSL_EOVRFLW when
> they can sense that l and m are too big."
>
> Roman.
>
> On Sep 14, 10:01 am, Bob Hanlon <hanl... at cox.net> wrote:
>
> > In version 6 I do not see a problem with the first two examples. In the third example, l is undefined. Defining l and changing 3.7 to 37/10 (use rational numbers to maintain high precision) works fine. Alternatively, specify higher precision (e.g., 3.7`25). Also, in version 6, Plot has an option to change (increase) the WorkingPrecision.
>
> > Bob Hanlon
>
> > ---- John Ralston <rals... at ku.edu> wrote:
>
> > > LegendreP[ l, m] and SphericalHarmonicY[ t, p, l, m]  go
> > > wrong for large index l .
>
> > > For l> 40 or so neither can be used reliably everywhere.
>
> > > To see the breakdown plot the functions.  Not all index cases fail.
> > > Here's some examples:
>
> > > Plot[LegendreP[ 44, 13, x] , {x, -1, 1} ]
>
> > > Plot[LegendreP[ 66, 9, x] , {x, -1, 1} ]
>
> > > ListPlot[ Table[ {j,
> > >      Sqrt[ 4Pi/ (
> > >       2l + 1) ] Abs[ SphericalHarmonicY[ j, 0, 0,
> > >                      3.7 ]]}, {j, 1, 55}], PlotJoined -> True]
> > ;
> > > Has anyone fixed this?  Does anyone care?
> > > I need l ranges above 200.
>
> > > thanks
> > > John Ralston
>
> > > > I find serious bugs in Mathematica 5.1 LegendreP
> > > > and SphericalHarmonicY.  It is not a matter of
> > > > definitions or syntax, but a catastrophic failure
> > > > easy to establish.  Math archives show a history
> > > > of discussion, but invariably centered on variations
> > > > of definition.
>
> > > > Wolfram Research shows little interest.  I'm
> > > > wondering if the failure
> > > > to perfom is well known and has been repaired by
> > > > someone
> > > > re-writing the commands.  I don't often follow this
> > > > forum, but I've joined to either get access to code
> > > > that works, or to inform people so they can go after
> > > > the problem.   Does anyone care?
>
> > > > John Ralston

```

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