[gmx-users] eigenvalues and number of frames

[Jose D Faraldo-Gomez]

Hi

Can anyone explain to me why when I use g_covar to build the covariance 
matrix and obtain the corresponding eigenvalues, the number of (clearly) 
non-zero eigenvalues appear to depend on the number of frames read (if this 
is smaller than 3N)?

E.g. if for a system of 134 C-alphas I do:

(a) g_covar_d -f $xtc -s $tpr -n $ndx -b 2000 -e 4000 -last 402 (200 frames)

then in eigenval.xvg I get

       (...)
       198 1.42414e-05
       199 1.40171e-05
       200 1.09781e-05
       201 -2.63089e-14
       202 2.55791e-14
       203 -9.5343e-15
       (...)

whereas if I do:

(b) g_covar_d -f $xtc -s $tpr -n $ndx -b 2000 -e 3000 -last 402 (100 frames)

I obtain

        (...)
        97 0.000174181
        98 0.000171367
        99 0.000164253
       100 0.000146225
       101 3.66036e-14
       102 -3.02553e-14
       103 -8.74524e-15
       (...)

And the same thing happens if I increase or decrease the number of frames 
being read by changing the sampling window length. In other words, the index 
of the last eigenvalue larger that, say, 1e-10, is exactly the same as the 
number of frames in the sampling window (until the number of frames exceeds 
the number of possible eigenvalues).

Also, if I tell g_covar to skip frames and do:

(c) g_covar_d -f $xtc -s $tpr -n $ndx -b 2000 -e 4000 -last 402 -dt 20 (100 
frames)

then I get:

        (...)
        97 0.000180717
        98 0.0001703
        99 0.00015828
       100 0.0001383
       101 -3.29179e-14
       102 3.07368e-14
       103 9.17597e-15
       (...)

which resembles the profile for the 1 ns window in (b), though of course the 
trace of the matrix is more similar to that in (a).

So, what am I missing? I can see how the covariance matrix changes as frames 
are included/excluded in the averaging, but once the matrix is built, should 
not be the number of non-zero eigenvalues independent of the number of 
frames? Apologies if this is really obvious...

Thanks in advance,

Jose


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