MATLAB: Ideas for vectorizing? (Computing multiple sumations simultaneously even though the max and min are different for each)

Question

I have two vectors, "nmin" and "nmax" which respectively contain the starting and ending indices for numel(nmin) sums that I need to compute. I would like to compute all of them simultaneously (i.e. vectorize). This is my current solution:

nmin = [3; 2; 16];
nmax = [5; 7; 16];
nsums = numel(nmin); %number of sums to be computed simultaneously
nn = nmax-nmin+1; %number of terms in the sum
maxnn = max(nn); %largest number of terms for any given sum
n = NaN(nsums,maxnn); %initialize as NaNs
for i = 1:nsums
    n(i,1:nn(i)) = nmin(i):nmax(i);
end
S = ((-1).^n)./n; %vectorized evaluation of each term in the sum
S(isnan(S)) = 0; %set the NaNs to zero now before performing the summation
S = sum(S,2); %perform the sums simultaneously

Basically, the idea is to create an array, "n" that contains all of the summation indices, but since each summation (rows of n) contains a different number of terms, I pad with NaNs:

n =
       3     4     5   NaN   NaN   NaN
       2     3     4     5     6     7
      16   NaN   NaN   NaN   NaN   NaN

In this example I have a sum from 3:5, another one from 2:7, and another one from 16:16 (i.e. only one term). After evaluating the terms, I set the NaNs to zero so that they don't contribute to the sum. For the example given above the result would be:

S =
     -0.2833
      0.2405
      0.0625

This works, but there are two performance problems:

(1) often times this algorithm produces a matrix "n" that has ~60% of its elements as NaNs, thus this is wasting memory and there are a lot of wasted computations when evaluating "S = ((-1).^n)./n;"

(2) if "nsums" is very large (and often it is on the order of 1e6) then I have a long for loop, which takes forever to evaluate.

Any ideas on how to avoid the for loop, or improve the performance in general?

Answer 1

 ub=max(nmax);
 n=1:ub;
 T=cumsum((-1).^(n)./n);
 S=T(nmax)-T(nmin-1)