Fixed - Polynomial Approximate Spectral Transformations for Preconditioning the Eigenvalue Problem

Fixed-Polynomial Approximate Spectral Transformations for Preconditioning the Eigenvalue Problem by Heidi K. Thornquist 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 wellseparated, 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 solving a linear system, which is computationally expensive for large-scale eigenvalue problems. This thesis shows how 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 fixedpolynomial operator with Arnoldi’s method are discussed. The computational results presented 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. Future research directions include development of convergence theory, accuracy bounds for computed eigenpairs, and alternative constructions of the fixed-polynomial operator.