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The main goals of these lectures are to introduce concepts of numerical methods and introduce MatLAB in an Engineering framework. By this we do not mean that every problem is a “real life” engineering application, but more that the engineering way of thinking is emphasized throughout the discussion. The philosophy of this book was formed over the course of many years. My father was a Civil Engineer and surveyor, and he introduced me to engineering ideas from an early age. At the University of Kentucky I took most of the basic Engineering courses while getting a Bachelor’s degree in Mathematics. Immediately afterward I completed a M.S. degree in Engineering Mechanics at Kentucky. While working on my Ph.D. in Mathematics at Georgia Tech I taught all of the introductory math courses for engineers. During my education, I observed that incorporation of computation in coursework had been extremely unfocused and poor. For instance during my college career I had to learn 8 different programming and markup languages on 4 different platforms plus numerous other software applications. There was almost no technical help provided in the courses and I wasted innumerable hours figuring out software on my own. A typical, but useless, inclusion of software has been (and still is in most calculus books) to set up a difficult ‘applied’ problem and then add the line “write a program to solve” or “use a computer algebra system to solve”. In these lecture notes, instruction on using MatLAB is dispersed through the material on numerical methods. In these lectures details about how to use MatLAB are detailed (but not verbose) and explicit. To teach programming, students are usually given examples of working programs and are asked to make modifications. The lectures are designed to be used in a computer classroom, but could be used in a lecture format with students doing computer exercises afterward. The lectures are divided into four Parts with a summary provided at the end of each Part. MatLAB and Solving Equations Vectors, Functions, and Plots in MatLAB MatLAB Programs Newton’s Method and Loops Controlling Error and Conditional Statements The Bisection Method and Locating Roots Secant Methods Symbolic Computations Linear Algebra Matrices and Matrix Operations in MatLAB Introduction to Linear Systems Some Facts About Linear Systems Accuracy, Condition Numbers and Pivoting LU Decomposition Nonlinear Systems - Newton’s Method Eigenvalues and Eigenvectors An Application of Eigenvectors: Vibrational Modes Numerical Methods for Eigenvalues The QR Method Iterative solution of linear systems Functions and Data Polynomial and Spline Interpolation Least Squares Fitting: Noisy Data Integration: Left, Right and Trapezoid Rules Integration: Midpoint and Simpson’s Rules Plotting Functions of Two Variables Double Integrals for Rectangles Double Integrals for Non-rectangles Gaussian Quadrature Numerical Differentiation The Main Sources of Error Differential Equations Reduction of Higher Order Equations to Systems Euler Methods Higher Order Methods Multi-step Methods ODE Boundary Value Problems and Finite Differences Finite Difference Method – Nonlinear ODE Parabolic PDEs - Explicit Method Solution Instability for the Explicit Method Implicit Methods Insulated Boundary Conditions Finite Difference Method for Elliptic PDEs Convection-Diffusion Equations Finite Elements Determining Internal Node Values