An efficient way to compute many sums with different number of terms

Suppose I have 2 functions and .

Given different values of , define

and

.

For each , I want to compute the sums (if ) or (if ) where and ϵ are given.

The number of terms varies across elements. For example,

I have then I need to compute 10 terms which are or then sum them up.

I have then I only need to compute 2 terms which are or then sum them up.

If the number of terms are all the same, i.e., , then I think I can vectorize and handle the computation. However, since the length of the sums are not the same, it becomes more difficult for me to handle it efficiently.

Can you help me to compute this in an efficient way?

Thank you very much in advance.

Answer 1

[Walter Roberson]

cumsum() and index it by the values

[Hung Dao]

Thank you for answering my question.

I think my question might be a little bit confusing so I have edited it.

Could you please be a little bit more specific on how to handle this using cumsum()?

[Walter Roberson]

Your revised explanation does not make sense to me unless a1 a2 and so on are functions?

[Hung Dao]

Yes. a1 a2 are functions of k and some other parameters, lets say x. They have the same formula. What makes they different is the values of x. For example, if the expression of the function is a = k*log(x), then a1 = k*log(x1) and a2 = k*log(x2)

[Walter Roberson]

Assuming that function a is vectorized in both inputs:

t = [t1, t2, t3, ...tn]
[K,T] = ndgrid(1:max(k_i), t);
A = a(K,T);
n = length(t);
Asum = sum(A .* ((1:n).' <= k_i(:).'), 1);

Likewise with b to create Bsum. Then you can do vectorized tests for which elements to select.

[Hung Dao]

Thank you very much. I implement your code and it works.

However, it might be very inefficient if there only a few that are very large (say 10,000 or more) and the other κ's are small. This is exactly the case which I am dealing with.

I am wondering whether a for loop might be more efficient in this case.

[Hung Dao]

In fact, what I am doing is to set then do the vectorization. I think it's quite similar to your suggestion. But the problem is that it is unnecessarily expensive. In principle, I can reduce the amount of computation by determining the optimal for each . However, this makes an efficient coding become much more challenging for me. I could use a for loop but it is the worst solution since I already have 2 for loops nested.

Is there any trick or Matlab function that I can use to handle this efficiently?