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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