For all current LAPACK symmetric eigensolvers, the error on the i-th eigenvalue
is proportional to machine precison times the largest (in magnitude)
eigenvalue. So the relative error might be large for small (in magnitude)
eigenvalues indeed. This is what people means with the statement
``larger eigenvalues are computed more stably than small eigenvalues''.
It might be better to say ``larger (in magnitude) eigenvalues are computed with
higher relative accuracy than small (in magnitude) eigenvalues''.
This is what is written in Section 4.7.1 of LAPACK Users' Guide in the formula:
- Code: Select all
| exact_lambda_i - computed_lambda_i | <= p(n) . epsilon . exact_lambda_max
Note: to get high relative accuracy for each eigenvalue (lambda_i), you would need
to have the exact_lambda_i on the right most side instead of exact_lambda_max.
Note: we have one singular value solver which enables high relative accuracy for
all singular values. This is a contribution from Zlatko Drmac and Kresimir
Veselic and this uses ``Jacobi SVD''. No such method for the symmetric