| Several applications require that we find a few eigenvalues of a large sparse matrix A∈Cnxn . These applications include structural analysis, stability of chemical reactions and the study of electrical systems. The sizes of these matrices are such that standard methods including the QR-algorithm generally require more memory than is available. Hence in 1992, Sorensen introduced the Implicitly Restarted Arnoldi process and in 1996, Sleijpen and van der Vorst introduced the Jacobi-Davidson process.; The goal of this dissertation is to develop methods which improve the convergence rate for these processes while reducing the work and memory requirements. The first main improvement is to alter how the initial space for the Jacobi-Davidson process is developed The Jacobi-Davidson process can be shown to be a Newton-Raphson scheme. Since Newton-Raphson schemes work much better if the initial guess is close to the desired answer, we have substituted linear methods such as subspace iterations at the start of the process to find reasonable approximations to begin the Jacobi-Davidson process with.; The second major alteration is to make adaptations to the Bi-Conjugate Gradient stabilized process for solving linear systems of equations. These adaptations allow for the substitution of the Bi-Conjugate Gradient stabilized process for the Generalized Minimum Residual process. This substitution reduces the memory requirements of the overall process to find the desired eigenvalues. Combining the first two alterations allow for successful use of the Jacobi-Davidson process in situations which were previously unsuccessful.; These alterations, along with several minor changes, makes it possible to increase the search space by several vectors during each step of the Jacobi-Davidson process. These vectors can be computed independently of each other and thus allows for parallel processing. |
