Convergence Rate Of Iterative Methods Research Articles

Alternating direction implicit preconditioning methods for self adjoint elliptic differential equations

In this paper, the application of preconditioning to improve the convergence rates of iterative methods for solving large sparse linear systems of finite difference equations arising from the discretisation of elliptic partial differential equations of self adjoint form on a rectangular grid is extended to the alternating direction implicit (A. D. I) methods. Theoretical results are presented for the model problem.

On the convergence rate of iterative methods for the solution of positive definite linear equations

The convergence rates of the point-Jacobi method and of the successive overrelaxation (including the Gauss-Seidel) method for symmetric positive definite linear equations are determined as functions of the spectrum of the matrix which is scaled to ones on the principal diagonal. It is seen that the convergence rate is strongly dependent upon the magnitude of the smallest eigenvalue and that for even mildly ill-conditioned matrices the convergence is very slow. It is also shown that the finite difference representation of the three-dimensional Laplace operator leads to remarkably well-conditioned matrices even when the number of equations is very large. Numerical experiments were performed supporting the analysis. It is concluded that the methods studied are virtually limited to large finite difference systems, but that they are very well adapted to such systems.