MATLAB: Double indexed recurrence relation

double indexrecurrence relationsequence

I need to simulate the following (double indexed!) recurrence relation:
p[1,1]=1
for n=1,2,3,… and k=1,2,…,n+1
p[n+1,k] = p[n,k]*(1-v*k/n) + p[n,k-1]*v*(k-1)/n , with p[n,0]=p[n,n+1]=0
where v is just some constant in the interval (0,1)
For a fixed value of n, say n=20, I need to compute p[n,k] for k=1,2,…,n
Thanks, Dave

Best Answer

I may be missing something but I don't understand the difficulty. Just use two loops:
maxn = 20;
v = rand;
p = zeros(maxn);
p(1,1) = 1;
for n = 1 : maxn-1
   for k = 1 : maxn  
         p(n+1, k) = p(n, k)*(1-v*k/n);
         if k > 1   %to account for p(~, 0) which is not valid index in matlab
              p(n+1, k) = p(n+1, k) + p(n, k-1)*v*(k-1)/n;
         end
   end
end
The rows of p are the values of n, the columns the values of k.
There may be a more efficient way of generating that matrix, without loops. I've not really looked into the details of your recurrence.
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