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Here
is symmetric and roughly models matrices from quantum
chemistry.
is the identity matrix. The main diagonal of
has
elements
and the off-diagonal elements of the
upper
triangular portion are selected randomly from the interval
. We consider computing the smallest eigenvalue of
using the diagonal
preconditioning of the original Davidson method. The eigenvalues
of
are
, 0.721, 1.73, 2.77,
, 101.58. So we let
. The eigenvalues of
are 0.0, 0.263, 0.387, 0.500,
, 2.01. The eigenvalue
of
is much better separated relative to the entire spectrum than is
of
. In fact, the gap ratio of
for
is
, while
the gap ratio for the eigenvalue
of
is
. With gap ratio
16 times larger, asymptotic convergence is very roughly four times
faster.
The next example looks at a matrix that is tougher to precondition. Some
results can be given similar to those for preconditioning linear
equations.
Susan Blackford
2000-11-20