Here is a copy of the reply to the message that you sent to the forum:
LAPACK gives you small backward errors, but not necessarily small
forward errors, that is the computed solution is the exact solution of a
closely neighbouring problem. If your problem is ill-conditioned then
the computed solution is likely to differ from the exact solution and
the difference depends upon the condition. Remember that LAPACK is
numerical software, not symbolic software.
A full discussion of accuracy and stability is given in Chapter 4 of the
LAPACK Users' Guide, and the particular case of the nonsymmetric
eigenvalue problem is discussed in Section 4.8.
For now, if you want high accuracy, or to use higher precision, then you
will need to use a symbolic package such as Mathematica, or Maple.
For the future, there are plans to allow the use of higher precision
arithmetic in LAPACK, but for now the routines use single, or double