normcdf: speed impact by vectorization

Hi,

I am used to the fact that replacing a for-loop with vectorization in Matlab yields speed improvements (or at least does not degrade speed). I was trying to optimize code that relies on normcdf computation and found that vectorization actually makes things worse - with almost 2x slow-down.

Here is the code I use to verify that (simplified version for the purpose of highlighting an issue, outputs are not stored):

a = -50:0.01:50;
total_cnt = 10000;
a_mod = repmat(a, 1, total_cnt);

fprintf(1, 'Calculating single long array\n');
tic;
normcdf(a_mod, 0, 1);
toc;

fprintf(1, 'Calculating short array with for loop\n');
tic;
for k = 1 : total_cnt
    normcdf(k*a, 0, 1);
end
toc;

In both instances I calculate normcdf for exactly the same number of values (~10^8) - when doing for-loop I even change the underlying values to avoid potential simply caching of calculations (not sure if Matlab does it), so that each iteration is brand new set of values. Here is an example output (2015 Macbook Pro, Matlab 2018a):

Calculating single long array
Elapsed time is 3.396627 seconds.
Calculating short array with for loop
Elapsed time is 1.450725 seconds.

It looks like I get faster computation in for-loop than with single line - is there a reason for this? I did not have time to experiment too much with it, but I tried reduce the size of a by 10x (a = -50:0.1:50) and then vectorized version became a little faster (0.35s vs 0.44s). I wonder if there is some threshold I am hitting?

Answer 1

[dpb]

Yeah, there is... --

>> a = -50:0.01:50;
>> total_cnt = 10000;
>> a_mod = repmat(a, 1, total_cnt);
>> whos a  a_mod
  Name      Size                        Bytes  Class     Attributes

  a          1x10001                    80008  double              
  a_mod      1x100010000            800080000  double              
>>

You created a really, really long vector and disk thrashing ensued...you may have more memory and so less than here but it took 150 sec on this machine...otoh

>> a = -50:0.01:50; a=a.';
>> a_mod = repmat(a, 1, total_cnt);
>> whos a_mod
  Name           Size                   Bytes  Class     Attributes

  a          10001x1                    80008  double              
  a_mod      10001x10000            800080000  double              

>> tic,normcdf(a_mod, 0, 1);toc
Elapsed time is 13.780391 seconds.
>>

won by about 11X over the single vector and 4X over the (probably intended) bunch of same length vectors.

Moral! Be sure you know what you're timing!!!

[Ilya Toytman]

so is the problem due to the fact that I don't have enough memory to hold long array, but enough memory to hold a matrix of the same number of elements? That seems counter-intuitive to me. Or is Matlab better optimized to work with 2D matrices than 1D arrays?

[dpb]

The vector is 10000X the size of the short one; you didn't store the array of 10000 columns but only the base vector. You would have to compare

a_moda=repmat(a,1,total_cnt);
a_modv=repmat(a.',1,total_cnt);

to compare that.

>> whos a*
  Name            Size                       Bytes  Class     Attributes

  a               1x10001                    80008  double              
  a_moda      10001x10000                800080000  double              
  a_modv          1x100010000            800080000  double              
  ans             1x1                            8  double              

>> 

[Ilya Toytman]

Yes, you are right. I just tried and I am getting slight improvement with 2D array but not that dramatic