| This dissertation is concerned with methods for solving systems of linear equations of the form Au = b, where A is a large sparse nonsingular matrix. When A is symmetric positive definite, the conjugate gradient method is often used and is fairly well understood. However, when A is nonsymmetric, the choice of iterative methods for solving the linear system is much more difficult.The GMRES method is considered to be a more stable method for solving nonsymmetric linear systems. However, the work per iteration increases as the number of iterations increases. Lanczos-type methods such as LANDIR, LANMIN(BCG), and LANRES require less work per iteration. However, they are usually less stable.In this dissertation, we consider the GGMRES method as well as two new methods, namely the MGMRES and LANMGMRES methods. The GGMRES method is a slight generalization of the GMRES method. Instead of using a minimization process as in GGMRES, we use a Galerkin condition to derive the MGMRES method. The LANMGMRES method is designed to combine the reliability of GMRES with the reduced work of the Lanczos-type methods.A computer program has been implemented based on the use of LANMGMRES for solving nonsymmetric linear systems arising from certain elliptic problems. Comparative numerical tests have been made with other available iterative methods for solving such problems. LANMGMRES have proven to be competitive with the other methods both in terms of iteration counts and also in terms of convergence behavior. |
