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

[Tomonori K. <ju...@qu...>]

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.