Iterative methods for the numerical solution of linear systems

The objective of this dissertation is the design and analysis of iterative methods for the numerical solution of large, sparse linear systems. This type of systems emerges from the discretization of Partial Differential Equations. Two special types of linear systems are studied. The first type deals with systems whose coefficient matrix is two cyclic whereas the second type studies the augmented linear systems. Initially, the Preconditioned Simultaneous Displacement (PSD) method, which is a generalized version of the Symmetric SOR (SSOR) method, is studied when the Jacobi iteration matrix is weakly cyclic and its eigenvalues are all real “real case” or all imaginary “imaginary case”. The first result is that the PSD method has better convergence rate than the SSOR method. In particular, in the “imaginary case” its convergence is increased by an order of magnitude compared to the SSOR method. In an attempt to further increase the convergence rate of the PSD method, more parameters were introduced. The new method is called the Modified PSD (MPSD) method. Under the same assumptions the convergence of the MPSD method is studied. It is shown that the optimum MPSD method is equivalent to the optimum MSOR method. Furthermore, the convergence analysis of the Generalized Modified Extrapolated SOR (GMESOR) and Generalized Modified Preconditioned Simultaneous Displacement (GMPSD) methods is studied for the numerical solution of the augmented linear systems. The main result of our analysis is that both methods possess the same rate of convergence and less complexity than the Preconditioned Conjugate Gradient (PCG) method. The last result is important since it proves that the addition of parameters in an iterative method has the same effect in the increase of the rate of convergence as that of the Conjugate Gradient (CG) method which belongs to the Krylov subspace methods.