الفهرس 
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Abstract
Summary
Thesis title: “Block Iterative Methods and Acceleration Techniques”.
The main objective in this work is based on the fact that the rate of
convergence of the block Jacobi method is approximately the same as that
of the point Gauss Seidel method. The aim of this thesis is to introduce a
new version of group iterative methods for the algebraic systems arising
from the discretization of boundary value problems in the plane. Some
special forms of a relatively new iterative block technique are introduced,
the KSLOR, KS2LOR, and GKSOR.
The thesis consists of four chapters, Arabic summary, and English
summary.
Chapter One: Basic Concepts
The basic concepts required in investigating iterative techniques for
solving linear systems are introduced in a simple form.
Chapter Two: Block Iterative Methods
Standard Block Iterative methods (Block Jacobi, Block Gauss-
Seidel, Block Successive over-relaxation) are studied, moreover the Block
KSOR method is introduced. The Block relaxation schemes are
generalizations of the point relaxation schemes. They update a whole set of
components of unknown vector at each time, typically a sub-vector of the
solution vector, instead of only one component. Special block iterative
methods are discussed, namely the Line iterative methods (Line Jacobi,
Line Gauss-Seidel, Line SOR, and Line KSOR). Also, the two Line
iterative methods with the natural ordering as well as the red black ordering
are discussed.
Chapter Three: group Iterative Methods
Different group iterative methods are considered, through the
treatment of Poisson equation defined on a unit square with 8 8 64
internal mesh points. The novel approach of using groups of fixed size is
considered. I.e. Groups of a particular number of individual equations
(mesh points), each group is dealt in the same way as a single point. Our
fundamental concern is to develop new grouping of the mesh points into
small size groups of 2, 4, 8 and 16 points and to investigate their
advantages as well as choosing the most efficient group for solving such
linear systems which arise from the discretization of PDE’s in two-space
dimensions.
Chapter Four: Preconditioned and Convergence Acceleration
Preconditioning is one of the most usable acceleration techniques.
The approach of preconditioning is demonstrated through simple forms of
well known types. It is proved that the preconditioning gives considerable
improvement in the rate of convergence for the iterative method (Jacobi,
Gauss-Seidel, SOR, and KSOR). It is demonstrated that the line SOR
method is a more efficient method than preconditioning treatment.
Moreover, the 2 line is more efficient than the line method.
• The objective of the thesis is to introduce a new versions of block
iterative methods.
• The motivation of the thesis is the Discretization techniques of
partial differential equations usually produce matrices which are
banded, or block banded.
It is worth to mention that:
- All calculations were done by using Matlab R2012b in the Mathematics
Department, Faculty of Science, Ain Shams University.
- The results of chapter two are published in the Journal of “Applied and
Computational Mathematics”.