# What are some ideas to preprocess / precondition the following linear system?

Let $A\in \mathbb{R}^{n\times n}$ symmetric and positive semidefinite, and $\omega\in \mathbb{R}\setminus\{0\}$. I am interested in solving the following linear system for a range of values of $\omega$:

$$((A-\omega^2 I)(A-\omega^2 I)+\omega^2 I)x = b.$$ It may be useful to note that the matrix factors as $$(A-(\omega^2-i\omega)I)(A-(\omega^2+i\omega)I),$$ where $i^2 = -1$.

Details: $A$ is sparse and I won't have direct access to its entries. The dimension of the null space of $A$ is a non-negligible fraction of $n$. The dimension of the problem, $n$, will be as big as the computer's RAM will allow.

What is a good way to solve preprocess / precondition this system? Note that the RHS, $b$, will change when $\omega$ changes.

Notes: This is a follow up question to this one. The idea of the proposed solution to that question shows that if we could perform an complete eigendecomposition on $A$, we would have a pretty much ideal preprocess. I have implemented an Lansczos iteration to approximate this eigendecomposition but it doesn't perform as well as I had hoped. I can explain this idea in more detail as an addendum if there is interest.

Of course full answers are appreciated, but they are not expected. I am mainly looking for ideas to investigate. Any comments and pointers to the literature are much appreciated.

Note to mods: Is this kind of question acceptable? I can change it to something more definite if asking for ideas is unacceptable.

# Edit

This is what I plan on doing. First note that as $\omega\to \infty$ the matrix starts looking like $I(\omega^4+\omega^2)$, so we are mainly interested in when $\omega$ is comparable to the norm of $A$ and smaller.

To that end, we compute $r$ eigen-pairs of $A$, $(\lambda_i,q_i)\in \mathbb{R}\times \mathbb{R}^{n\times n}$, with the largest eigenvalues. Then, since these eigenvectors can be made to be orthonormal we have $$x= \sum_{i=1}^r \alpha_i q_i + \sum_{i={r+1}}^n \alpha_i q_i.$$

Now, taking the dot product of both side of the equation with $q_i$ for $1\le i\le r$ we get

$$\alpha_i = \left\langle q_i,b \right\rangle \frac{1}{(\lambda_i - \omega^2)(\lambda_i - \omega^2) + \omega^2}.$$

I plan on using this information to construct an initial guess for $x$. I am still unsure on what preconditioner to use.

• I think it's fine (although I'm not a mod); the main thing to keep in mind is that it should be (somewhat) clear from the question what constitutes a useful answer, and ideally, that there is some objective (or at least not purely subjective) way of choosing the "best" answer to accept, in case there are multiple. Oct 16 '15 at 15:36

You have noticed that any eigenvector of $A$ is also an eigenvector of your composed matrix. When you compute eigenvectors of your matrix, you can use them in a deflation-type preconditioner, as described in [1, 2].