Arnoldi's method is often used to compute a few eigenvalues and eigenvectors of large, sparse matrices. When the eigenvalues of interest are not dominant or well-separated, this method may suffer from slow convergence. Spectral transformations are a common acceleration technique that address this issue by introducing a modified eigenvalue problem that is easier to solve than the original. This modified problem accentuates the eigenvalues of interest, but requires a linear solve, which is computationally expensive for large-scale eigenvalue problems.

We will show this expense can be reduced through a preconditioning scheme that uses a fixed-polynomial operator to approximate the spectral transformation. Implementation details and accuracy heuristics for employing a fixed-polynomial operator with Arnoldi's method will be discussed. Computational results will also be presented, which indicate that this preconditioning scheme is a promising approach for solving large-scale eigenvalue problems. Furthermore, this approach extends the domain of applications for current Arnoldi-based software.