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The Comparison of Gaussian Elimination and Cholesky Decomposition Methods to Linear System of Equations

The Comparison of Gaussian Elimination and Cholesky Decomposition Methods to Linear System of Equations

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DEDICATION

This research material, titled “The Comparison of Gaussian Elimination and Cholesky Decomposition Methods to Linear System of Equations” is dedicated to God for His boundless grace and guidance. It is also a tribute to all computer enthusiasts whose contributions made my research journey smoother and enriched my documentation process, making the experience truly fulfilling.



ACKNOWLEDGEMENT

I am profoundly grateful to everyone who contributed to the successful completion of this project. I am especially grateful to my Supervisor (Name), the Head of Department (Name), and the Lecturers in the Department of Physics for their invaluable guidance and support. I also acknowledge the contributions of authors and scholars whose works on The Comparison of Gaussian Elimination and Cholesky Decomposition Methods to Linear System of Equations provided essential insights. Special thanks go to my study area (and any funding organizations, if applicable) for their financial assistance. I am equally thankful to stakeholders, including mentors, teachers, and colleagues, for their encouragement and support. Finally, I deeply appreciate my family and friends for their patience and unwavering support throughout this journey. Your contributions have been instrumental in making this research a reality.



TABLE OF CONTENTS

PRELIMINARY PAGES

CHAPTER ONE

INTRODUCTION

  • 1.1 Introduction
  • 1.2 Background of Study
  • 1.3 Aim and Objectives of Study
  • 1.4 Significance of Study
  • 1.5 Scope of Study
  • 1.6 Limitations of the Study
  • 1.7 Definition of Terms

CHAPTER TWO

LITERATURE REVIEW

  • 2.1 Introduction
  • 2.2 Gaussian Elimination Method
  • 2.4 Checking Procedures in Gaussian Elimination Method
  • 2.5 Advantages of Gaussian Elimination Method
  • 2.6 Disadvantage of Gaussian Elimination Method
  • 2.7 Solving Linear Systems
  • 2.8 Methods of Solving Linear System of Equation

CHAPTER THREE

CHOLESKY DECOMPOSITION METHOD

  • 3.1 Introduction
  • 3.2 Derivation of Cholesky Method on a Case by Case Base
  • 3.3 Advantages of Cholesky Decomposition Method
  • 3.4 Disadvantages of Cholesky Decomposition Method

CHAPTER FOUR

RESULTS AND DISCUSSION

  • 4.1 Introduction
  • 4.2 Application of Linear System
  • 4.3 Comparison between Gaussian Elimination Method and Cholesky Decomposition Methods

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATION

  • 5.1 Introduction
  • 5.2 Summary of Findings
  • 5.3 Conclusion
  • 5.4 Recommendation

REFERENCES


ABSTRACT

Guassian elimination is a tool for obtaining the solution of equations, to compute the determinant, for deducing rank of coefficient matrix. It is the standard method for solving linear equations. The study was conducted to compare the Linear System of Equations using Gaussian Elimination and Cholesky Decomposition Methods. In achieving this aim, the following specific objectives were laid out to ascertain the difference between Gaussian Elimination and Cholesky Decomposition Methods, determine the easiest methods of solving linear programming problems and examine the linear programming problem using Gaussian Elimination and Cholesky Decomposition Methods. 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 indept 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. This study will be of immense benefit to Statisticians, Mathematicians and other researchers who intend to know more on this study and can also be used by non-researchers to build more on their research work. This study contributes to knowledge and could serve as a guide for other study.


The Comparison of Gaussian Elimination and Cholesky Decomposition Methods to Linear System of Equations

CHAPTER ONE

1.1 Introduction

Guassian elimination is the standard method for solving linear equations. As it is a ubiquitous algorithm and plays a fundamental role in scientific computation. Guassian elimination is a tool for obtaining the solution of equations, to compute the determinant, for deducing rank of coefficient matrix. However Guassian Elimination depends more on matrix analysis and computation it emphasis on block pivoting, methods of iteration and a means to improve the computed solution quality. It involves two stages forward and backward stage. Forward stage: Unknowns are eliminated in this stage by manipulation of equations and constitute an echelon form. Backward stage: it is related with back substitution process on the reduced upper triangular method resulting in a solution of equation.

As a prelude to other parts of this study, this chapter will discuss the background upon which this study was initiated, the statement of problems that led to this study, the Aim and Objectives of the study. Others are Significance of the study, Scope of work, Limitations of the Study and Definition of technical terms.

…

CHAPTER TWO

2.0 Literature Review

2.1 Introduction

This chapter focuses on the review of related literature. A literature review includes the current knowledge as well as theoretical and methodological contributions to a particular topic. It documents the state of the art with respect to the topic you are writing. It surveys the literature in the topic selected. In this research work the literature review includes the conceputal review, theoretical framework, the review of related literature …

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