normalized ALF (Assotiated Legendre Function)

[Ben Abbott]

On Feb 12, 2008, at 3:06 AM, Marco Caliari wrote:

>> I compared Matlab's script and Octave's and your proposal.
>>
>> result_matlab = legendre (80, [-1:0.1:1]);
>>
>> result_octave0 respects the original and result_octave1 respects  
>> your algorithm.
>>
>> I calculated an error with respect to the matlab version (I'm not  
>> sure Matlab's is to be trusted as correct in all cases).
>>
>> d0 = abs (result_matlab - result_octave0) / abs (result_matlab);
>> d1 = abs (result_matlab - result_octave1) / abs (result_matlab);
>> er1 = max (d1, [], 2);
>> er0 = max (d0, [], 2);
>>
>> [er0(:), er1(:)] produces the following
>
> Dear Ben,
>
> the results are even more impressive if you consider that  
> legendre0(80,[-1:0.1:1])(1,1) gives 6.7015e+14, whereas legendre1(80, 
> [-1:0.1:1])(1,1) gives 1.
>
>> I'm inclined to agree that the recursion form should work better.  
>> I'm suspicious that Matlab's version is reliable for such high  
>> order legendre polynomials.
>>
>> Anyone, is there a reliable method to verify the correct answers?
>
> There is a LegendreP function in Maple doing exactly the same. Can  
> you select few cases (a degree and a scalar value for x) for which  
> the difference between Matlab and my script is quite large? I will  
> check them with Maple.

I haven't used Maple in several years. Is it not possible to evaluate  
the example, legendre (80, [-1:0.1:1]),  and post the results (as an  
attachment)?

It can then read it as ...

	result_maple = load ("result_maple.txt");

and compared directly to octave and Matlab. I realize that we'll loose  
some precision, but if 16 digits are preserved, we should obtain  
relative errors on the order 10^(-15).

I notice Maxima also has a legendre function, legendre_p(n,m,x). Which  
uses `Abramowitz and Stegun, Handbook of Mathematical Functions' as  
the reference. I'll take a look at calculating the example there.

Ben