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

[Maho N. <ch...@ma...>]

Hi Gang,

From: Gang Yan <ee...@gm...>
Subject: Re: [Mplapack-devel] Can I compute eigenvalues of ill-conditioned matrix less than 10^-156 ?
Date: Fri, 23 Sep 2011 00:55:35 +0800

>   but the smallest eigenvalues as pointed out ~~~~~~~~~ are negative. So it
> is wrong. I think the program can only compute the eigenvalues which are
> larger than 1e-156 accurately. Do you think so? Or I am wrong for some
> reasons? Thanks.

in your case, you use 

> //initialization of MPFR
>    int default_prec = 4096;
>    mpfr_set_default_prec(default_prec);

then, machine epsilon is approximately 7.458340731e-155 (*).
In this case, relative errors of 
eigenvalues can be less accurate than 7.458340731e-155.
you are doing something complicated in your program and
I guess some rouding errors included. In my opinion,
therefore, you found some small negative eigenvalues.

(*) you can check this by inserting following code and rerun your
program.
    mpreal a = Rlamch_mpfr("E");
    mpfr_out_str(stdout, 10, 10, a, MPFR_RNDN);
    printf("\n");
Thanks,
-- Nakata Maho http://accc.riken.jp/maho/ , JA OOO http://ja.openoffice.org/
http://blog.goo.ne.jp/nakatamaho/ ,GPG: http://accc.riken.jp/maho/maho.pgp.txt