Re: [Mplapack-devel] Can I compute eigenvalues of ill-conditioned matrix less than 10^-156 ?

[Gang Y. <ee...@gm...>]

Hi,  Tomonori,

 Thank you very much. Yes, for very ill-conditioned matrices, eigenvalues
are very sensitive to round-off errors when computing with float-point. I
think I can obtain the smallest eigenvalues if using higher precision
arithmetic.

best,
gang

On Mon, Sep 12, 2011 at 11:01 PM, Tomonori Kouya <ju...@qu...>wrote:

> Hi, Gang. I have tried to solve some eigenvalue problems using MPFR/GMP.
>
> Very ill-conditionded matrices such as Hilbert or Lotkin matrices, are
> too sensitive to keep their theoretical properties due to initial error
> or round-off error in computing process. In case of Hilbert matrix, it
> is theoretically positive definite but some of smallest eigenvalues can
> be minus due to initial error in approximated elements of Hilbert matrix.
>
> I recommend to know how sensitive your matrix is for such error by using
> variable precision of MPFR/GMP in order to solve your troubles.
>
> (2011/09/12 23:35), Gang Yan wrote:
> >     Now I have a problem when I calculate the eigenvalues of an extremely
> > ill-conditioned matrix, in a problem of control system research. The
> matrix
> > is not very large, e.g., 30X30, but extremely ill-conditioned, i.e.,
> having
> > the largest eigenvalue ~10^-2 and the smallest eigenvalue>  10^-156. I
> found
> > that the eigenvalue programme using GMP or MPFR can only compute
> eigenvalues
> > <10^-156, or I will get some minus eigenvalues which is not wrong because
> > the matrix is positive-definition. I use a 64-bit notebook.
>
>
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