# A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems
## 7 Numerical Results
### Iterative Eigensolvers
In order to define a more effective preconditioner for shifted linear systems one can allow the preconditioned matrix \(PA\) to have eigenvalues different from one corresponding to the columns of matrix \(W\). In Reference [50] a generalized block tuned (GBT) preconditioner is defined:  
**Definition 3**. : _Given a preconditioner \(P_{0}\), an \(n\times p\) matrix \(W\) with full column rank, and a diagonal matrix \(\Gamma=\text{diag}(\gamma_{1},\ldots,\gamma_{p})\), a generalized block tuned preconditioner for matrix \(A\) is a matrix \(P\) obtained by correcting \(P_{0}\) with a low-rank matrix depending on \(A,W\) and \(\Gamma\) and satisfying_  
\[PAW=W\Gamma. \tag{18}\]  
The generalized SR1 preconditioner is defined as  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]0  
Note that the above preconditioner is not in general symmetric as small matrix \(\Pi\) is not and hence its use would prevent convergence either of the DACG eigensolver or the inner PCG iteration within the Newton method. However, this drawback can be circumvented when \(W\equiv W_{j}\) represents the matrix of the (approximate) eigenvectors corresponding to the diagonal matrix with the eigenvalues of \(A\): \(\Lambda_{j}=\text{diag}(\tilde{\lambda}_{j+1},\tilde{\lambda}_{j+2},\ldots, \tilde{\lambda}_{m})\). In such case we can approximate \(\Pi\) as  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]1  
so restoring symmetry. This modified preconditioner \(\widetilde{P}_{j}=P_{0}-Z\tilde{\Pi}^{-1}Z^{T}\) satisfies only approximately the tuning property as \[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]2  
The following theorem states that it is however possible to have \(p\) eigenvalues of the preconditioned matrix \(\widetilde{P}_{j}(A-\theta_{j}I)\) very close to one depending on how the columns of matrix \(W\) approximate the eigenvectors of \(A\). We also define the reciprocal of the relative separation between pairs of eigenvalues as  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]3  
**Theorem 3**. : _Let matrix \(W_{j}\) be as in (17), \(\widetilde{P}_{j}\) an approximate GBT preconditioner satisfying condition (21), with \(\gamma_{s}=\bar{\lambda}_{s}/(\bar{\lambda}_{s}-\bar{\lambda}_{j}),\;s=j+1, \dots,m\), then each column of \(W_{j}\) is an approximate eigenvector of \(\widetilde{P}_{j}(A-\theta_{j}I)\) corresponding to the approximate eigenvalue_  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]4  
_satisfying_  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]5  
Proof. : See Reference [50]. The eigenvalues \(\mu_{s}\) can be very close to one depending on the initial tolerance \(\tau\) as stated in Corollary 1, under reasonable additional hypotheses  
**Corollary 1**. : _Let \(\theta_{j},\tau\) such that, for all \(j=1,\dots,m-1\):_  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]6 \[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]7  
_then_  
\[J_{k}^{(j)}\mathbf{s} = -(A-\theta_{j}^{(k)}I)\mathbf{u}_{k};\quad\text{for}\quad\mathbf{ s}\perp\mathbf{u}_{k},\qquad\text{where} \tag{13}\]8  
Proof. : See Reference [50]. From Corollary 1 it is clear that \(\mu_{s}\) can be made arbitrarily close to one by appropriately reducing the tolerance \(\tau\). As an example, if \(\xi_{j}=10^{2}\), \(\tau=10^{-3}\) then all \(\mu_{s}\) are expected to be in \((1,1.2)\).