ABSTRACT
This project work is concerned with study of the comparison of Gaussian elimination and cholesky decomposition methods to linear system of equations.
In chapter one, we are concerned with linear systems and the various methods of solving them. In chapter two and chapter three, we dealt with the indepth study of Gaussian Elimination method and the cholesky Decomposition method, with their good points and bad points respectively. While chapter four deals with various application area of both methods, their comparison and finally conclusion.
TABLE OF CONTENT
ABSTRACT
TABLE OF CONTENT
CHAPTER ONE: LINEAR SYSTEM OF EQUATIONS
1.1INTRODUCTION
1.2SOLVING LINEAR SYSTEM
1.3METHODS OF SOLVING LINEAR SYSTEM
CHAPTER TWO: GAUSSIAN ELIMINATION METHOD
2.1 INTRODUCTION
2.2 DERIVATION OF GAUSSIAN ELIMINATION METHOD
2.3 CHECKING PROCEDURES IN GAUSSIAN ELIMINATION
2.4 ADVANTAGES OF GAUSSIAN ELIMINATION METHOD
2.5 DISADVANTAGES OF GAUSSIAN ELIMINATION METHOD.
CHAPTER THREE: CHOLESKY DECOMPOSITION METHOD
3.1 INTRODUCTION
3.2 DERIVATION OF CHOLESKY DECOMPOSITION METHOD ON A CASE BY CASE BASE
3.3 ADVANTAGES OF CHOLESKY DECOMPOSITION METHOD
3.4 DISADVANTAGES OF CHOLESKY DECOMPOSITION METHOD.
CHAPTER FOUR:
4.1 INTRODUCTION
4.2 APPLICATION OF LINEAR SYSTEMS
4.3 COMPARISON BETWEEN GAUSSIAN ELIMINATION AND CHOLESKY DECOMPOSITION METHODS
4.4 CONCLUSION
4.5 REFERENCES.
CHAPTER ONE
LINEAR SYSTEM OFEQUATIONS
1.1 INTRODUCTION
Therehave been series of method used in solving systems of linear equations. Asystem of equation is a set or collection of equations solved together (NoreenJamil, 2012). Collection of linear equations is termed as system of linearequations. They are often based on same set of variables. Various methods havebeen evolved to solve the linear equations but there is no best method yetproposed for solving system of linear equations.
Among other method of solving linear system ofequations, the Guassian method and the choleskey composition methods will beapplied discussed in details.
A wide variety of problems lead ultimately to the needto solve a linear system of equation linear system of equations are associatedwith many problems in engineering and science as well as with applications ofmathematics to the social sciences and the quantitative study of business andeconomic problems.
In 1985, according to Atkinson, system of Simultaneouslinear equation occur in solving problems in a wide variety of areas withrespect to mathematics, statistics, physical quantities (examples aretemperature, voltage, population management and displacement). Social sciences,engineering and business. They arise directly in solving real life problems.
The world sometimes reveals itself to us as observablerelationships among the relevant variables what it does make evident are relationshipthat describe how both the variable and their rate of change affecteach other.
Apparently, such life changing problem gives rise tosystems of simultaneous linear equation. In almost every human activities, manseems to be compelled to uncover fundamental relationship that exist among the objects he observes.According to Maron in 1982, he said in order to make the relationship that existbetween variables explicit, we frequently attempt to make a mathematical modelthat will accurately reflect real life situation. Many mathematical models thatwill accurately reflect real life situation. Many mathematical models have thesame basic structure although disparity in Symbolic rotation may be utilized,which can arise from economics, transportation, which need may arise to makeefficient allocation among several points or to solve the growth of populationin which units of x1, x2 …., xn arises fromnet flow from one point to another or in relationship to population growth,that is, number of individuals in a particular age group at a particular time.
There are various methods in solving linear system ofsimultaneous equations. In numerical analysis the techniques and methods forsolving system of linear equations belongs to two categories: Direct andIterative methods. The direct methods obtain the exact solution (in realarithmetic) in finitely many operations where as iterative method generate asequence of approximations that only converge in the limit to the solution. Thedirect method falls into two categories or clam that is the Gaussianelimination method and cholesky decomposition method. Some others are matrixinverse method and LU factorization method and the Cramer’s rule method.
The elimination approach reduces the given system ofequations to a form from which the solution can be obtained by simplesubstitution since calculators and computers have some limit to the number ofdigits for their use this may lead to round-off errors and produces poorerresults. Generally, the direct method are best for full or bounded matriceswhere as iterative methods are best for very large and sparse matrices. Theiterative method provide an alternative to the direct methods for solvingsystems of linear equations. This method involves assumption of some initialvalues which are then refined repeatedly till they reach some accepter rang ofaccuracy. The Jacobi and Gawn-siedel methods are good examples of the iterativemethod.