Abstract: Analysis of real life problems often results in linear
systems of equations for which solutions are sought. The method to
employ depends, to some extent, on the properties of the coefficient
matrix. It is not always feasible to solve linear systems of equations
by direct methods, as such the need to use an iterative method
becomes imperative. Before an iterative method can be employed
to solve a linear system of equations there must be a guaranty that
the process of solution will converge. This guaranty, which must
be determined apriori, involve the use of some criterion expressible
in terms of the entries of the coefficient matrix. It is, therefore,
logical that the convergence criterion should depend implicitly on the
algebraic structure of such a method. However, in deference to this
view is the practice of conducting convergence analysis for Gauss-
Seidel iteration on a criterion formulated based on the algebraic
structure of Jacobi iteration. To remedy this anomaly, the Gauss-
Seidel iteration was studied for its algebraic structure and contrary
to the usual assumption, it was discovered that some property of the
iteration matrix of Gauss-Seidel method is only diagonally dominant
in its first row while the other rows do not satisfy diagonal dominance.
With the aid of this structure we herein fashion out an improved
version of Gauss-Seidel iteration with the prospect of enhancing
convergence and robustness of the method. A numerical section is
included to demonstrate the validity of the theoretical results obtained
for the improved Gauss-Seidel method.
Abstract: We study the semiconvergence of Gauss-Seidel iterative
methods for the least squares solution of minimal norm of rank
deficient linear systems of equations. Necessary and sufficient conditions
for the semiconvergence of the Gauss-Seidel iterative method
are given. We also show that if the linear system of equations is
consistent, then the proposed methods with a zero vector as an initial
guess converge in one iteration. Some numerical results are given to
illustrate the theoretical results.
Abstract: A generalized Dirichlet to Neumann map is
one of the main aspects characterizing a recently introduced
method for analyzing linear elliptic PDEs, through which it
became possible to couple known and unknown components
of the solution on the boundary of the domain without
solving on its interior. For its numerical solution, a well conditioned
quadratically convergent sine-Collocation method
was developed, which yielded a linear system of equations
with the diagonal blocks of its associated coefficient matrix
being point diagonal. This structural property, among others,
initiated interest for the employment of iterative methods for
its solution. In this work we present a conclusive numerical
study for the behavior of classical (Jacobi and Gauss-Seidel)
and Krylov subspace (GMRES and Bi-CGSTAB) iterative
methods when they are applied for the solution of the Dirichlet
to Neumann map associated with the Laplace-s equation
on regular polygons with the same boundary conditions on
all edges.
Abstract: In this article, the phenomenon of nonlinear
consolidation in saturated and homogeneous clay layer is studied.
Considering time-varied drainage model, the excess pore water
pressure in the layer depth is calculated. The Generalized Differential
Quadrature (GDQ) method is used for the modeling and numerical
analysis. For the purpose of analysis, first the domain of independent
variables (i.e., time and clay layer depth) is discretized by the
Chebyshev-Gauss-Lobatto series and then the nonlinear system of
equations obtained from the GDQ method is solved by means of the
Newton-Raphson approach. The obtained results indicate that the
Generalized Differential Quadrature method, in addition to being
simple to apply, enjoys a very high accuracy in the calculation of
excess pore water pressure.