       Re: LegendreP error (bug?) in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg81275] Re: LegendreP error (bug?) in Mathematica
• From: Yaroslav Bulatov <yaroslavvb at gmail.com>
• Date: Tue, 18 Sep 2007 00:41:25 -0400 (EDT)
• References: <fcdf41\$pjg\$1@smc.vnet.net><fclajn\$f4a\$1@smc.vnet.net>

```I agree with Roman that Mathematica's implementation of LegendreP is
suboptimal. Consider the precision of native LegendreP[n,x]
implementation vs 3-term recursion formula as a function of n

legendre3term[0, x] = 1;
legendre3term[1, x] = x;
legendre3term[n_, x_] := Module[{p1 = 1, p2 = x},
For[i = 2, i < n + 1, i++,
{p1, p2} = {p2, ((2*i - 1)*x*p2 - (i - 1)*p1)/i};
];
p2
];
ints = Block[{\$MinPrecision = 50, \$MaxPrecision = 50},
x = N[Interval[1/10], \$MinPrecision];
{legendre3term[#, x], LegendreP[#, x]} & /@ Range[1, 100]
] /. {Interval[{a_, b_}] :> b - a};
ListLinePlot[Transpose[-Log[10, ints]],
PlotLabel ->
"Precision of LegendreP and 3-term recursion for large n",
AxesLabel -> {"n", "Precision"}]

in this example, native LegendreP implementation loses .55 digits of
precision each time n increases by 1, while 3-term recursion loses
0.041
legendre3term also runs faster than LegendreP for a few values I tried

Yaroslav

On Sep 17, 12:33 am, Roman <rschm... at gmail.com> wrote:
> 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-...
> 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 := Series[LegendreP[200, 43, x], {x, 4/5, 1}] // Normal // N
> > Out = 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 := LegendreP[200, 43, 0.8`50]
> > Out =
> > 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 := LegendreP[200, 43, 0.8] // Precision
> > Out = 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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