Unexpected numerical errors in matrix/vector multiplication

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José Goulart on 4 Sep 2014
Commented: José Goulart on 8 Sep 2014
Hi,
I have observed an apparently very strange behavior when running a Matlab script. I have a very long data vector t (with 4560000 elements) and several other vectors x1,...,xM, all with the same dimension as t. Now, I basically want to compute the scalar product between each xm and t. I have done that using two procedures: the first one consists in computing each scalar product between a vector xm and t inside a for loop, while the other simply consists in forming a matrix X whose columns are the vectors xm and then computing the product of X transposed and t. Surprisingly, the results are very different!
For simplicity, let us suppose M=2. If I compute:
>> y1 = [x1 x2].'*t;
>> y2 = zeros(2,1);
>> y2(1) = x1.'*t;
>> y2(2) = x2.'*t;
Then I get a large relative normalized error
>> er = norm(y2-y1)/norm(y1)
er =
6.3080e-05
(By the way, norm(y1) = 1.2665e+06). Componentwise, the error is
>> y1(1)-y2(1)
ans =
1.5918
>> y1(2)-y2(2)
ans =
-0.0898
In principle, these alternatives should always yield the same results, regardless of the contents of those vectors, since they should be translated into machine code corresponding to equivalent operations. So, it seems odd to me that some kind of numerical issue arises.
Has anyone seen a similar behavior before and knows what might be causing that? Am I missing something with respect to the internal implementation of such operations?
Thanks,
Henrique
arich82 on 4 Sep 2014
I get something really, really weird.
If I start a new session of Matlab, the first time I run the code, I get no difference, but the second (and subsequent) time I run the code, I get non-trivial errors:
>> t=rand(1000,1);
x=rand(2,1000);
z1=x*t;
z2=zeros(2,1);
z2(1)=x(1,:)*t;
z2(2)=x(2,:)*t;
z1-z2
ans =
0
0
>> t=rand(1000,1);
x=rand(2,1000);
z1=x*t;
z2=zeros(2,1);
z2(1)=x(1,:)*t;
z2(2)=x(2,:)*t;
z1-z2
ans =
1.0e-12 *
-0.1990
0.0568
If I restart Matlab, same thing happens. This is in 8.3.0.532 (R2014a) running in Ubuntu 14.04, with dual Intel E5-2690 v2.

Oleg Komarov on 4 Sep 2014
Edited: Oleg Komarov on 4 Sep 2014
The dot product implementation might differ according to the input, i.e. dot product between two vectors as opposed between two matrices. Also, a quick search on the web exposes a few implementations of the dot product in the BLAS routines themselves and Matlab might be picking one or another according to different needs.
The example suggests zdotu.f
edit([matlabroot '/extern/examples/refbook/dotProductComplex.c']);
However, this doesn't tell us much about the dot product between matrices.
The difference in the results after startup calculation seems to be just chance. In fact, if you vary the number of random draws, the difference is not 0 anymore. To reset the state of the random generator without restarting, simply use
rng('default');
To address such a big discrepancy, we need to see what are the values of the dot products.
José Goulart on 8 Sep 2014
Indeed, that must be the cause.
Also, since the elements of the vectors span several orders of magnitude, adding them in different orders should produce fairly different results.
Thanks, everyone.