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

[Douglas Bates]

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.