I have a square matrix of the form: $$ \begin{pmatrix} a & b \\ -b & a \end{pmatrix} $$ where $$ a = \begin{pmatrix} D1 & t & 0 & 0 & t & 0 \ldots t \\ t&D2&t&0&0&t\ldots \\ 0&t&D3&t&0&0 \ldots \\ 0&0&t&D4&t&0 \ldots \\ t&0&0&t&D5&t\ldots\\ 0&t&0&0&t&D6\ldots\\\ldots\\\ldots\\ t\ldots \end{pmatrix} $$ is a laplacian (symmetric) matrix where there is also a top right element at $(1,N)$ and an equal bottom left element at $(N,1)$ also equal to $t$.
The matrix $b$ is diagonal, but could become more complex in the future (probably also Laplacian).
Since the total matrix is by construction non symetric I get real and complex eigenvalues for small dimensions. I was wondering if there is an easy way to obtain one or more eigenvalues at high dimensions. I have tried with ARPACK (by default used in Matlab) which uses Arnoldi iteration with implicit restarts. It fails to find eigenvalues to sufficient accuracy though. The cause seems to be that my matrices have a relatively high condition number. If it's possible, maybe could you suggest what algorithm would be better suited (or should I maybe modify a bit) based on the structure of the matrix? Any suggestions is welcome