Example 11.2.1.

Here $A$ is symmetric and roughly models matrices from quantum chemistry. $B$ is the identity matrix. The main diagonal of $A$ has elements $1, 2, 3, \ldots, 100$ and the off-diagonal elements of the upper triangular portion are selected randomly from the interval $(-1,1)$. We consider computing the smallest eigenvalue of $A$ using the diagonal preconditioning of the original Davidson method. The eigenvalues of $A$ are $-0.323$, 0.721, 1.73, 2.77, $\ldots$, 101.58. So we let $\lambda_1 = -0.323$. The eigenvalues of $T_{\mathrm{IC}} = (D-\lambda_1 I)^{-1}(A-\lambda_1 I)$ are 0.0, 0.263, 0.387, 0.500, $\ldots$, 2.01. The eigenvalue $0$ of $T_{\mathrm{IC}}$ is much better separated relative to the entire spectrum than is $\lambda_1$ of $A$. In fact, the gap ratio of $\lambda_1$ for $A$ is ${{\lambda_2 - \lambda_1} \over {\lambda_n-\lambda_2}} = 0.0094$, while the gap ratio for the eigenvalue $0.0$ of $T_{\mathrm{IC}}$ is $0.151$. 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.