# A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems
## 4 Choice of the Vectors \(\{\mathbf{w}_{j}\}\)  
In principle every set of linearly independent vectors can be selected as columns of \(W\). However, the optimal choice is represented by the eigenvectors of the **preconditioned matrix**\(P_{0}A\) corresponding to the smallest eigenvalues. Approximation of these vectors as a by-product of the Conjugate Gradient method has been considered in various works. We will turn on this crucial issue later on.  
We start by presenting some numerical results obtained computing the \"true\" 10 leftmost eigenvectors of \(P_{0}A\) using the Matlab eigs command. To measure the sensitivity of the method to eigenvector accuracy we also use perturbed eigenvectors that is, satisfying \(\|P_{0}A\mathbf{w}_{j}-\mu_{j}\mathbf{w}_{j}\|\approx\delta\). The coefficient matrix was chosen as the FD discretization of the Laplacian on an L-shaped domain obtained using the Matlab command  
A = delsq(numgrid('L',500)); which returns a sparse matrix of order \(n=186003\). The linear system \(A\mathbf{x}=\mathbf{b}\), with \(\mathbf{b}\) a random uniformly distributed vector, was solved by the PCG method with various low-rank update techniques with \(P_{0}=IC(0)\).  
The results in Table 4 show that a very low accuracy on the eigenvectors (absolute residual of order of \(\delta=0.01\)) is sufficient to provide good preconditioning results. Also notice that the tuned preconditioner seems to be more sensitive to the parameter \(\delta\).  
It may also happen that some of the leftmost eigenpairs of the **coefficient matrix** are instead available. What if one uses these vectors as columns of \(W\)? The following Table 5 gives a surprising answer: if they are computed to a good accuracy they provide a notable acceleration of convergence. Again we use either exact eigenvectors or vectors satisfying \(\|A\mathbf{w}_{j}-\lambda_{j}\mathbf{w}_{j}\|\approx\delta\).  
The conclusion is: yes we can use the leftmost eigenvectors of \(A\) instead of those of \(P_{0}A\) without dramatically loosing efficiency. The reason for this behavior is connected to the action of preconditioners such as Incomplete Cholesky or approximate inverses. They usually leave almost unchanged the eigenvectors corresponding to the smallest eigenvalues (though increasing the latter ones).  
Consider again the previous matrix and compare the eigenpairs \((\lambda_{j},\mathbf{v}_{j})\) of \(A\), with those of the preconditioned matrix with two choices of \(P_{0}\): \(IC(0)\) (\(\mu_{j}^{P1},\mathbf{v}_{j}^{P1}\)) and the IC factorization with \(\mathtt{droptol}=1e-2\) (\(\mu_{j}^{P2},\mathbf{v}_{j}^{P2}\)). As expected, in Figure 1 (left) we give evidence that the preconditioners shift bottom-up the smallest eigenvalues. However the angle between the corresponding eigenvectors, computed as  
\[\angle\left(\mathbf{u},\mathbf{v}\right)=\arccos\frac{\mathbf{u}^{T}\mathbf{v }}{\|\mathbf{u}\|\cdot\|\mathbf{v}\|},\]  
and displayed in Figure 1, right, turns out to be close to zero showing that they are almost parallel.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **No Update** & **Tuned** & **Deflated** & **Spectral** \\ \hline exact & 466 & 254 & 254 & 254 \\ \(\delta=0.01\) & 466 & 261 & 259 & 290 \\ \(\delta=0.05\) & 466 & 378 & 260 & 286 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Influence of accuracy of the eigenvectors of \(P_{0}A\) on the efficiency of the low-rank preconditioners. PCG iterations with various updating strategies are reported.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **No Update** & **Tuned** & **Deflated** & **Spectral** \\ \hline exact & 466 & 254 & 254 & 254 \\ \(\delta=10^{-3}\) & 466 & 296 & 296 & 297 \\ \(\delta=0.01\) & 466 & 362 & 361 & 369 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Influence of accuracy of the eigenvectors of \(A\) on the efficiency of the low-rank preconditioners. PCG iterations with various updating strategies are reported.