# A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems
## 7 Numerical Results
### Iterative Eigensolvers  
When computing the smallest or a number of the interior eigenvalues of a large and sparse matrix, most iterative solvers require the solution of a shifted linear system of the type  
\[(A-\sigma B)\mathbf{x}=\mathbf{b}.\]  
This is true for instance in the shift-invert Arnoldi method, in the inverse power iteration, in the Rayleigh Quotient iteration. A related linear system is also to be solved within the _correction equation_ in the Jacobi-Davidson method. Other gradient-based methods such as the Locally Optimal Block PCG method (LOBPCG) [46] or the Deflation-Accelerated Conjugate Gradient method (DACG) [47] implicitly solve a shifted linear system. The sequence of linear systems are characterized by a constant or a slowly varying parameter \(\sigma\) so informations on matrix \(A\) or on the pencil \((A,B)\) can be profitably used for all the sequence. We denote as \(0<\lambda_{1}\leq\lambda_{2}\leq\ldots\lambda_{m}\ldots\leq\lambda_{n}\) its eigenvalues and \(\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{m},\ldots\mathbf{v}_{n}\) the corresponding eigenvectors. Our aim is to investigate the efficiency of the SR1 tuned preconditioner in the acceleration of the so-called correction equation in the simplified Jacobi-Davidson (JD) method [48; 49]. To assess eigenpair \((\lambda_{j},\mathbf{v}_{j})\) a number of linear systems have to be solved of the form  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\] \[J_{k}^{(j)} = (I-QQ^{T})(A-\theta_{j}^{(k)}I)(I-QQ^{T}),\,\theta_{j}^{(k)}=q( \mathbf{u}_{k})\equiv\frac{\mathbf{u}_{k}^{T}A\mathbf{u}_{k}}{\mathbf{u}_{k}^ {T}\mathbf{u}_{k}},\] (14) \[Q = \left[\mathbf{v}_{1}\quad\ldots\mathbf{v}_{j-1}\quad\mathbf{u}_{ k}\right].\]  
Figure 4: Number of iterations for each linear system in the sequence and various preconditioning strategies. Initial preconditioner: IC (\(A_{1},1e-5\)) (upper plots) and IC (\(A_{2},1e-4\)) lower plots. The PCG method is proved to converge if applied to systems (13) as the Jacobian \(J_{k}^{(j)}\) is shown to be SPD in the subspace orthogonal to \(\mathbf{u}_{k}\)[22; 49]. Following the works in [27; 50], which are based on a two-stage algorithm, the columns of \(W\) are orthonormal _approximate_ eigenvectors of \(A\) provided by a gradient method, DACG [47], satisfying  
\[A\mathbf{\hat{v}}_{s}=\tilde{\lambda}_{s}\mathbf{\hat{v}}_{s}+ \mathbf{res}_{s},\qquad\|\mathbf{res}_{s}\| \leq \tau\lambda_{s}\qquad\quad s=1,\ldots,m. \tag{15}\] \[\tilde{\lambda}_{s}-\lambda_{s} \leq \tau\lambda_{s}. \tag{16}\]  
These vectors are also used as initial guesses for the Newton phase. To update a given initial preconditioner for \((\lambda_{j},\mathbf{v}_{j})\) we use the next approximate eigenvectors \(\mathbf{v}_{j+1},\ldots,\mathbf{v}_{m}\) to define an SR1 tuned preconditioner as  
\[P_{j}=P_{0}-Z\left(Z^{T}AW_{j}\right)^{-1}Z^{T},\quad Z=P_{0}AW_{j}-W_{j}, \text{ with }W_{j}=\left[\mathbf{\hat{v}}_{j+1}\quad\ldots\quad\mathbf{\hat{v}}_{m} \right]. \tag{17}\]  
We recall the following result stated in ([27], Lemma 3.1):  
**Lemma 1**. : _Let \(P_{j}\) a block tuned preconditioner satisfying condition (17). In the hypothesis (15) each column of \(W_{j}\) that is, \(\mathbf{\hat{v}}_{s},s=j+1,\ldots,m\), is an approximate eigenvector of \(P_{j}(A-\theta_{j}I)\) satisfying_  
\[P_{j}(A-\theta_{j}I)\mathbf{\hat{v}}_{s}=\left(1-\frac{\theta_{j}}{\tilde{ \lambda}_{s}}\right)\mathbf{\hat{v}}_{s}+\epsilon_{s},\qquad\text{with} \qquad\|\epsilon_{s}\|\leq\tau\lambda_{j+1}\|P_{j}\|.\]  
The effect of applying a tuned preconditioner to \(A-\theta_{j}I\) is to set a number of eigenvalues of \(P(A-\theta_{j}I)\) to a value that is close to one only under the conditions that the eigenvalues are well separated, that is, \(\dfrac{\lambda_{j}}{\lambda_{j+1}}\ll 1\), which is not always the case in realistic problems.