How to calculate local running total in a updating loop?

I have a variable A in a time updating loop, A is updated by a functoin f to another value:

While t<=T
A=A+f(B);
end

Now, Compute the running total of A is very simple:

S=0;
While t<=T
A=A+f(B);
S=S+A;
end

However, I only want the 10 most recent values of A to be added up in S:

S(t)=A(t-9)+A(t-8)+A(t-7)+A(t-6)+A(t-5)+A(t-4)+A(t-3)+A(t-2)+A(t-1)+A(t);

How could I do this in the loop? Note that it is not practical to save all the histroy of A into the memory because A is extremely big, not even the 10 most recent values. What is the most memory efficient way to do this?

Thanks!

[Hao Zhang]

Any answers?