Matlab vectorization of multiple embedded for loops

Question

Suppose you have 5 vectors: v_1, v_2, v_3, v_4 and v_5. These vectors each contain a range of values from a minimum to a maximum. So for example:

v_1 = minimum_value:step:maximum_value;

Each of these vectors uses the same step size but has a different minimum and maximum value. Thus they are each of a different length.

A function F(v_1, v_2, v_3, v_4, v_5) is dependant on these vectors and can use any combination of the elements within them. (Apologies for the poor explanation). I am trying to find the maximum value of F and record the values which resulted in it. My current approach has been to use multiple embedded for loops as shown to work out the function for every combination of the vectors elements:

% Set the temp value to a small value
temp  = 0;

% For every combination of the five vectors use the equation. If the result
% is greater than the one calculated previously, store it along with the values 
% (postitions) of elements within the vectors
for a=1:length(v_1)
    for b=1:length(v_2)
        for c=1:length(v_3)
            for d=1:length(v_4)
                for e=1:length(v_5)
                     % The function is a combination of trigonometrics, summations, 
                     % multiplications etc..
                     Result = F(v_1(a), v_2(b), v_3(c), v_4(d), v_5(e))   
                     % If the value of Result is greater that the previous value, 
                     % store it and record the values of 'a','b','c','d' and 'e'
                     if Result > temp;
                          temp = Result;
                          f    = a;
                          g    = b;
                          h    = c;
                          i    = d;
                          j    = e;
                     end
                end
            end
        end
    end
end

This gets incredibly slow, for small step sizes. If there are around 100 elements in each vector the number of combinations is around 100*100*100*100*100. This is a problem as I need small step values to get a suitably converged answer.

I was wondering if it was possible to speed this up using Vectorization, or any other method. I was also looking at generating the combinations prior to the calculation but this seemed even slower than my current method. I haven't used Matlab for a long time but just looking at the number of embedded for loops makes me think that this can definitely be sped up. Thank you for the suggestions.

Answer 1

No matter how you generate your parameter combination, you will end up calling your function F 100^5 times. The easiest solution would be to use parfor instead in order to exploit multi-core calculation. If you do that, you should store the calculation results and find the maximum after the loop, because your current approach would not be thread-safe.

Having said that and not knowing anything about your actual problem, I would advise you to implement a more structured approach, like first finding a coarse solution with a bigger step size and narrowing it down successivley by reducing the min/max values of your parameter intervals. What you have currently is the absolute brute-force method which will never be very effective.