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.
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                             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.
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                                                  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
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