[Scilab-users] Bessel functions

[grivet]

Le 26/01/2015 10:57, Dang, Christophe a écrit :
> Hello,
>
>> De  Claus Futtrup
>> Envoyé : samedi 24 janvier 2015 17:53
>>
>> I've made a small script to play around with Bessel functions of the first kind
>> [...]
>> for m=0:19 // ...
>> powerseries_m = ((-1)^m / (factorial(m) * factorial(m + n))) * (x/2)^(2*m);
>> powerseries = powerseries + powerseries_m; // sum the powerseries
>> end
> Generally speaking, I think you'd better use the possibilities of vector calculation as much as possible, to have a faster calculation.
>
> The lines above could look like the following, assuming x is a column vector:
>
> m = 0:19;
>
> [x_mat, m_mat] = ndgrid(x, m); // rectangular matrices for group evaluation
>
> f = 1 ./(factorial(m) .* factorial(m + n));
>
> f_mat = ndgrid(f, x)';
>
> powerseries = sum((-1).^m_mat.*f_mat.*(x_mat/2).^(2*m_mat), "c");
>
> However, this quite naive implementation is probably not accurate when x and m are high,
> for f would have some zeros (if m or m+n > 170) leading to zeros in powerseries,
> whereas f*(x/2)^(2m) might not be negligible.
>
> The reason why you should follow Nikolay's advice and use built-in functions, which usually use strong algorithms.
>
> --
> Christophe Dang Ngoc Chan
> Mechanical calculation engineer
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Almost every numerical analysis textbook comprises a chapter on the 
evaluation of Bessel functions, as for instance
     Abramowitz ans Stegun, Handbook of mathematical functions, chapters 
9,10 (elegant use of recurrence relation)
     Press et al., Numerical recipes, chapter 6 (rational approximations)
Enjoy!