Optimizing MATLAB code

Question

This code takes an extremely long time to run (more than 10 minutes). Is there any way in which I can optimize it so that it finishes in less than one minute?

Answer 1

Vectorizing your algorithm where I could reduced the execution time slightly to ~8.5 minutes. Calculate all of the harmonic sums in one statement:

harmonicsum = cumsum(1 ./ (1:1e6));

You can now calculate the right-hand side in one statement:

rhs = harmonicsum + log(harmonicsum) .* exp(harmonicsum);

I couldn't vectorize the determination of the factors so this is the fastest way I could come up with to sum them. MATLAB's FACTOR command allows you to generate all the prime factors for each iteration. We then compute the unique set of products of all possible combinations using UNIQUE and NCHOOSEK. This avoids testing each integer as a factor.

lhs = zeros(1e6, 1);
for ii = 1:1e6
    primeFactor = factor(ii);
    numFactor = length(primeFactor);
    allFactor = [];
    for jj = 1:numFactor-1
       allFactor = [allFactor; unique(prod(nchoosek(primeFactor, jj), 2))];
    end
    lhs(ii) = sum(allFactor) + 1 + ii;
end
lhs(1) = 1;

Find the indices at which the Riemann hypothesis is violated:

isViolated = find(lhs > rhs);