Re: LegendreP error (bug?) in Mathematica

*To*: mathgroup at smc.vnet.net*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 MathLink. If you're not familiar with this, just ask. 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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**Re: LegendreP error (bug?) in Mathematica**

**Re: LegendreP error (bug?) in Mathematica**