# A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems
## 7 Numerical Results
### Iterative Eigensolvers
In Reference [50] (Theorem 2) is also stated that the eigenvalues of the preconditioned projected Jacobian matrix (13) are characterized in the same way as stated in Theorem 3, that is, that for a suitable function \(C(\tau)\), increasing in \(\tau\)  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]9  
where \(\widetilde{P}_{j}\) a generalized block tuned preconditioner, \(P_{Q}=(I-QQ^{T})\widetilde{P}_{j}(I-QQ^{T})\). To limit the cost of the application of the low-rank correction it is customary to fix the maximum number of columns of matrix \(W_{j}\), parameter \(l_{\max}\). Conversely, it is required to enlarge it when assessing eigenpairs with \(j\approx m\). The first DACG stage is hence used to compute an extra number (win) of approximated eigenpairs. In this way the number of columns of \(W_{j}\) is \(m_{j}=\min\{l_{\max},m+\text{win}-j\}\). The construction (C) of \(\widetilde{P}_{j}\) and its application (A) as \(\widetilde{P}_{j}\tau\) are sketched below. MVP = matrix-vector products, \(Z_{j}=Z_{0}(:\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!We report from Reference [50] the results of the described preconditioner in the computation of the 20 smallest eigenpairs of matrix thermomec, which is available in the SuiteSparse Matrix Collection at [[https://sparse.tamu.edu/](https://sparse.tamu.edu/)]([https://sparse.tamu.edu/](https://sparse.tamu.edu/)). This is a challenging test due to the high clustering of its smallest eigenvalues. The CPU times (in seconds) refer to running a Fortran 90 code on a 2 x Intel Xeon CPU E5645 at 2.40 GHz (six core) and with 4 GB RAM for each core. The exit test is on the relative eigenresidual: \(\dfrac{\|A\bu-q(\bu)\bu\|}{q(\bu)}\leq\varepsilon=10^{-8}\), with \(q(\bu)\) defined in (14). The results of the runs are displayed in Table 7 where the number of iterations (and CPU time) of the initial DACG method are reported as well as the cumulative number of iterations (and CPU time) needed for solving all the linear systems (13) within the Newton iteration for the 20 eigenpairs. Especially the second (Newton) phase takes great advantage by the GBT preconditioner as also accounted for by Figure 5 shows the number of iterations taken by the Newton phase with the fixed IC, SR1 tuned and GBT preconditioners, together the (scaled) \(\log\xi_{j}\), which is a measure of the condition number of the inner linear systems. The almost constant GBT curve confirms the property of this preconditioner which makes the number of iterations nearly independent on the relative separation between consecutive eigenvalues. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Phase**} & \multicolumn{3}{c}{**DACG**} & \multicolumn{3}{c}{**Newton**} & \multicolumn{3}{c}{**Total**} \\ \cline{2-10}  & **win** & \(\mathbf{l_{max}}\) & \(\mathbf{\tau}\) & **Its.** & **CPU** & **OUT** & **Inner** & & **MVP** & **CPU** \\ \hline Fixed & 0 & 0 & \(10^{-4}\) & 1510 & 15.86 & 153 & 2628 & 34.12 & 4291 & 52.97 \\ SR1 Tuned & 10 & 10 & \(10^{-3}\) & 1335 & 14.89 & 137 & 2187 & 32.19 & 3659 & 51.48 \\ GBT & 5 & 10 & \(10^{-3}\) & _777_ & 11.16 & 44 & 607 & 9.42 & **1428** & **20.74** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Timings and iterations for the DACG -Newton method for the computation of \(m=20\) eigenpairs of matrix thermomec. In boldface the smallest overall number of iterations and CPU time. Figure 5: Number of iterations for the Newton phase with fixed, SR1 tuned and generalized block tuned (GBT) preconditioners. In red the (scaled) logarithm of the indicator \(\xi_{j}\).