[R] How to set the floating point precision beyond e-22?

[Spencer Graves]

	  And don't forget to rescale first, as Peter Dalgaard suggested.

	  I've been giving this lecture for over 20 years, and I got caught by 
it myself a few years ago.  I computed the crossproduct matrix, as Doug 
said, and had no end of trouble because it was numerically singular.  I 
rescaled each column by subtracting some number roughly close to the 
mean or median and dividing by something close to the standard 
deviation, and the problem went away.

	  spencer graves

Douglas Bates wrote:

> On 8/5/05, Roy Nitze <rnitze at wiwi.uni-bielefeld.de> wrote:
> 
>>We have a problem inverting a matrix which has the following eigenvalues:
>>
>>
>>>eigen(tcross, only.values=TRUE)
>>
>>$values
>> [1]  7.917775e+20  2.130980e+16  7.961620e+13  8.241041e+12  2.258325e+12
>> [6]  3.869428e+11  6.791041e+10  2.485352e+09  9.863098e+08  9.819373e+05
>>[11]  3.263408e+05  2.929853e+05  2.920419e+05  2.714355e+05  8.733435e+04
>>[16]  8.127136e+04  6.543883e+04  5.335074e+04  3.773311e+04  2.904373e+04
>>[21]  2.418297e+04  1.387422e+04  8.925090e+03  5.538344e+03  4.831908e+03
>>[26]  1.133571e+03  9.882477e+02  7.725812e+02  5.081682e+02  3.010545e+02
>>[31]  1.801611e+02  1.319787e+02  1.050521e+02  7.096471e+01  5.576549e+01
>>[36]  4.192645e+01  3.549810e+01  2.638731e+01  2.444429e+01  1.735139e+01
>>[41]  1.058796e+01  7.425778e+00  7.209576e+00  4.689665e+00  3.181650e+00
>>[46]  3.002956e+00  1.959247e+00  1.551665e+00  1.079589e+00  1.064981e+00
>>[51]  5.409617e-01  4.076501e-01  2.010129e-01  1.302394e-01  4.029787e-02
>>[56]  2.599448e-02  1.061294e-02  1.634286e-03  4.095303e-09  1.021885e-10
>>[61]  2.124763e-11  6.906665e-12  2.850103e-12  9.440867e-13  6.269723e-13
>>[66]  1.043794e-13 -1.300171e-13 -7.220665e-13 -4.166945e-12 -6.145350e-12
>>[71] -2.776804e-11 -5.269669e-11 -7.154246e-10 -1.490515e-09 -1.294256e-08
>>[76] -1.224821e-02 -3.278657e+00 -4.620100e+01 -9.781843e+02 -1.303929e+04
>>[81] -5.545949e+04 -8.077540e+04 -8.577861e+04 -1.329961e+05 -1.450908e+05
>>[86] -3.022353e+05 -4.015776e+05
>>
>>As yout can see, the eigenvalues spread very much (between e+20 and e-13).
>>We presume, that it has something to do with R's floating point precision,
>>which I read is about 22-digits in mantissa as default. Can this precision
>>be set to values above 22? The problem occurs especially when trying to
>>perform 2SLS with the 'systemfit' package. There appears always an error
>>message like the following from the inverting routine:
>>
>>solve(tcross)
>>Error in solve.default(tcross) : Lapack routine dgesv: system is exactly
>>singular
>>
>>Or is there another source of error? We would like to embed R-routines in a
>>non-commercial web application for which we need to employ 'systemfit'
>>together with individual, user-submitted data. So the problem needs a
>>general solution and not a special one for this particular matrix.
> 
> 
> It appears that you are using the crossproduct but not taking
> advantage of the symmetry.
> 
> The condition number of the crossproduct is the square of the
> condition number of the original matrix so you are making your
> conditioning problems much worse by taking the crossproduct.  I
> suggest that you use a QR or SVD decomposition of the original model
> matrix instead.  You will still end up with a very ill-conditioned
> problem but now quite as bad as the one you have now.
> 
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