Double indexed recurrence relation

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

[Dave]

I have looked around online, but can't find anyway to deal with the double index in matlab

Answer 1

[Guillaume]

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.

[Dave]

Thank you very much Guillaume!