performance - Matlab - How to sum 1 to 1E10 serie of 1/x optimaly (in a few minutes) - matlab slows down

Question

I have a problem with matlab. I need to sum math serie 1/x from 1 to 1E10. I have some code in Matlab which contains loop - first loop step is ok very fast), but on second step in loop it slows down and Matlab is almost freezed, so I cannot calculate this in appropriate time.

Can you help me with this?

For smaller range it works OK (for example 1E06), but I need to calculate for the whole range. I have tried to separate to smaller range, but there is still loop and matlab is very slowed.

It looks like a problem with matlab and for loop, which slows down. After first loop step the RAM is full but for the second loop step the RAM is still full so it slows down. I don′t know why the Matlab does not free the RAM.

Thank you for any help!

Vladimir

Answer 1

You want to obtain the 1e10-th harmonic number. In the Symbolic Toolbox there is a function for that, called harmonic, and it's very fast:

>> format long %// to see more decimals
>> n = 1e10;
>> harmonic(n)
ans =
  23.603066594891992

The reason why it's so fast is that the harmonic function exploits the relationship between harmonic numbers, the Euler-Mascheroni constant and the digamma function:

where "psi" is the digamma function, H_n is the n-th harmonic number, and "gamma" is the Euler-Mascheroni constant. So you could also use

>> n = 1e10;
>> vpa(psi(n+1) + eulergamma)
ans =
    23.603066594891987434787570068504

If you don't have the Symbolic Toolbox, you can still do:

>> g = 0.5772156649015328606065120900824; %// Euler-Mascheroni constant
>> n = 1e10;
>> psi(n+1) + g
ans =
  23.603066594891988