[Scilab-users] Bernoulli numbers calculation

[Samuel Gougeon]

Hello Lester,


Le 29/12/2021 à 09:00, Lester Anderson a écrit :
> Hello,
>
> A quick query. Have adapted existing Matlab code for Scilan to 
> calculate Bernoulli numbers using an explicit formula 
> (www.bernoulli.org <http://www.bernoulli.org>).

Not so explicit. Could you please provide the formula, before 
implementing it in Scilab language?
The following is a "vectorized" version of your code, likely with the 
same mistake:

B=[];
for  m  =  0:20
     Sum  =  0;
     for  k=0:m
         v  =  0:k;
         Sum  =  Sum  +  sum(((-1).^v).*nchoosek(k,v).*(v.^m)/(k+1));
     end
     B(m+1)  =  Sum;
end

A working implementation based on the recurrent formula 
Bm=−1m+1∑k=0m−1(m+1k)Bk

could be:

mMax  =  20;
B  =  zeros(1,mMax);
B(1:2)  =  [1  -1/2];
for  m  =  2:2:mMax
     B(1,m+1)  =  -sum(nchoosek(m+1,0:m-1).*B(1,1:m))/(m+1);
end--> B ans  =         column 1 to 13   1.  -0.5   0.1666667   0.  
-0.0333333   0.   0.0238095   0.  -0.0333333   0.   0.0757576   0.  
-0.2531136


          column 14 to 21
    0.   1.1666667   0.  -7.0921569   0.   54.971178   0. -529.12424

To fasten the loop, the call to nchoosek() could be replaced with the 3 
last rows of Pascal's triangle.

Regards
Samuel
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