الفهرس 
Chapter 1 Rectangular and Singular Linear SystemsOnly 14 pages are availabe for public view
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Abstract
Summary
Thesis title: “Iterative Methods for Solving Singular and Rectangular Systems”
In general solving systems of linear equations is one of the most important problems in mathematics. Due to the progress in mathematical modelling of realistic problems and their treatments in addition to the difficulties in solving such models exactly, numerical methods becomes the appropriate choice. Systems of linear equations appears in approximately all real models directly or when using approximate methods in solving such models. Non-standard systems of equations appears as a result of the developments in modelling real life problems. Large rectangular linear systems appear in the statistical treatment of data and the problems associated with data fitting by using least squares concepts. Singular linear systems appear in the numerical solutions of differential equations with periodic characters. Recently, iterative techniques for solving large systems of algebraic equations become the common approach due to the progress in computational systems. Using iterative techniques even with standard systems require some rearrangements of the equations to guarantee the convergence of the technique, the situation become more complicated with rectangular and singular systems. Our main objective in this work is the study of rectangular and singular systems and their possible rearrangements to become suitable for using successive iterative techniques. Efficient use of the successive over relaxation method (SOR) requires a good choice of the relaxation parameter. We discussed the choice of the relaxation parameter and we introduced the KSOR versions of the SOR method for the augmented systems appears in the use of the concept of least squares.
This thesis consists of four chapters as follows:
Chapter One: Rectangular and Singular Linear Systems
The basic concepts of rectangular and singular systems are introduced. The generalized invers and their algorithms are used with application to a numerical example. The concept of singular value decomposition is discussed with an algorithm for its calculation with application to a numerical example. The least squares approach is discussed
with its use in solving linear algebraic systems. The Poisson equation are considered with different boundary conditions (Dirichlet, Neumann and periodic boundary conditions). Poisson’s equation are transformed to algebraic system form using the five point difference formula.
Chapter Two: Rectangular Linear Systems and Iterative Techniques
An overdetermined linear system is considered with full rank coefficient matrix, 𝐴 𝑋=𝑏, 𝐴∈ 𝑅𝑚×𝑛, rank(A) = n. The system is reformulated to be suitable for the use of iterative techniques. The 3-block KSOR and the 2-block KSOR versions of the corresponding SOR are introduced. The least squares solution to overdetermined linear system is obtained by using iterative methods (3-block SOR, 3-block KSOR, 2-block SOR and 2-block KSOR). Comparison of the performance of the iterative methods is considered. Discussion of the selection of the relaxation parameters are discussed.
Chapter Three: Singular Linear Systems and Iterative Techniques
The convergence and the semi convergence of iterative methods are established. The optimal iterative methods (SOR, KSOR, and JOR) for solving singular and nonsingular linear systems arising from the numerical treatment of Poisson equation in two dimensions are considered. The iterative methods (SOR, KSOR, and JOR) are applied in the algebraic singular and nonsingular linear systems. The relations for optimum relaxation parameters are established. A comparison between convergence rates are studied.
Chapter Four: Rank Deficient 4 Block and Preconditions
The 4-block SOR method is introduced for the rank deficient overdetermined linear system, the convergence and optimal convergence factors are introduced. The treatment of overdetermined linear system are considered as well as applying the iterative methods (SOR and KSOR) on the normal equation. Two types of preconditioned matrices are applied to the system of normal equations. The comparison between the convergence rates for the all methods are established