Review of Elementary Numerical Methods and MATLAB®

Summary In this chapter, we review the use of MATLAB® in the context of several elementary applications from ordinary differential equations and calculus. First, we practice basic matrix and vector operations at the command line. Next, we write a short code to implement Euler’s method, which we then easily modify to implement the Runge–Kutta method for systems. We recall the shooting method, an ODE-only technique for solving BVP via repeated application of an IVP solver. We further demonstrate MATLAB syntax and programming techniques on several elementary numerical differentiation and integration problems, apply Newton’s method to nonlinear systems, and solve a cubic spline with a single linear system solve step. The included homework exercises require the reader to implement the example programs, and introduce some new concepts. Each section ends with one or more subsections containing a brief discussion of the various aspects of MATLAB which one needs in order to implement and execute the algorithms. This material attempts to address the many frequently asked questions concerning the nuts and bolts of getting a program to produce appropriate output. We introduce some concepts and techniques for the comparison of approximate solutions to exact solutions, a pedagogical theme repeated in the HW exercises throughout this textbook. We present in gray boxes overviews of specific MATLAB commands and syntax used in these programs which might be new to the uninitiated, and which are relied on heavily throughout this text.