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Numerical Methods for Solving Systems of Nonlinear Equations by Courtney Remani Aproject submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301 (Honour’s Seminar) Lakehead University Thunder Bay, Ontario, Canada c copyright (2012-2013) Courtney Remani Abstract This Honours Seminar Project will focus on the numerical methods involved in solv- ing systems of nonlinear equations. First, we will study Newton’s method for solving multivariable nonlinear equations, which involves using the Jacobian matrix. Second, we will examine a Quasi-Newton which is called Broyden’s method; this method has been described as a generalization of the Secant Method. And third, to s solve for nonlin- ear boundary value problems for ordinary differential equations, we will study the Finite Difference method. We will also give an application of Newton’s method and the Finite Difference method. Using the computer program Matlab, we will solve a boundary value problem of a nonlinear ordinary differential system. i Acknowledgements I would like to acknowledge and thank everyone who has been involved with this project. I would like to express my sincere gratitude to my supervisor Dr. Liping Liu. Without her knowledge, direction, guidance, and all of her help, this project would not have been achieveable. I would also like to show a great deal of appreciation to my project coordinator, Dr. Adam Van Tuyl. He has been a great professor to me over the years. I would also like to acknowledge how thankful I am for my Mom and Dad. Without their unconditional love and support, and always believing in me throughout the years, I would not have been able to achieve a lot of my goals here at Lakehead University. ii Contents Abstract i Acknowledgements ii Chapter 1. Introduction 2 Chapter 2. Preliminaries 3 Chapter 3. Newton’s Method 7 Chapter 4. Broyden’s Method 15 Chapter 5. Finite-Difference Method 18 Chapter 6. Matlab Application 24 Chapter 7. Conclusion 29 Appendix 31 Bibliography 35 iii
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