Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs $$(\lambda ,x)$$(λ,x) of $$(\lambda ^2M+\lambda C+K)x=0$$(λ2M+λC+K)x=0 with M, C, K symmetric $$n\times n$$n×n matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization $$Ay=\lambda By$$Ay=λBy, where A, B are symmetric $$2n\times 2n$$2n×2n matrices but the pair (A, B) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices M, C, K as in the Q-Arnoldi method. This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.