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MathGroup Archive 2000

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LegendreP & Gauss quad bug

  • To: mathgroup at smc.vnet.net
  • Subject: [mg25414] LegendreP & Gauss quad bug
  • From: Paul Cally <cally at kronos.maths.monash.edu.au>
  • Date: Fri, 29 Sep 2000 01:07:11 -0400 (EDT)
  • Organization: Monash University
  • Sender: owner-wri-mathgroup at wolfram.com

There is a bug in the V4 implementation of the Legendre function
LegendreP, at
least in the linux version.

I am running Mathematica 4.0 under RedHat 6.2 linux. I have found the
following bug:

Mathematica 4.0 for Linux
Copyright 1988-1999 Wolfram Research, Inc.
 -- Motif graphics initialized --

In[1]:= LegendreP[5,2,N[1/3]]

Out[1]= -10.3704

In[2]:= LegendreP[5,2,N[1/3,20]]

Out[2]= -0.17283950617283951

In[3]:= %%/%

Out[3]= 60.

===================================

LegendreP[n,m,x] appears to work properly if x is a machine precision
number, but if it is higher precision, it gives a result which is out by

EXACTLY the factor m!/n! This error is NOT present in Version 3.0
running
under Digital Unix.

Please note that this also causes the standard package
GaussianQuadratures
to give wrong results for the quadrature weights if n is odd and
precision
is set to greater than 16.

In[1]:= << NumericalMath`GaussianQuadrature`

In[2]:= GaussianQuadratureWeights[3, -1, 1]

Out[2]= {{-0.774597, 0.555556}, {0, 0.888889}, {0.774597, 0.555556}}

In[3]:= GaussianQuadratureWeights[3, -1, 1,17]

Out[3]= {{-0.77459666924148337703585, 0.98360655737704918033},

>    {0, 0.0327868852459016393443},

>    {0.77459666924148337703585, 0.98360655737704918033}}

If n is even, there is no abscissa at x=0, and the error scales away.
However, if n is odd, the unscaled weight associated with the exact
number
x=0 is correct but all the others are wrong. When normalized, all the
weights are therefore greatly in error.


Paul Cally



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|Assoc Prof Paul Cally            |    Ph:  +61 3 9905-4471                |
|Dept of Mathematics & Statistics |    Fax: +61 3 9905-3867                |
|Monash University                |    paul.cally at sci.monash.edu.au        |
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|AUSTRALIA                        | http://www.maths.monash.edu.au/~cally/ |
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