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Section1.1An Introduction to These IBL Notes

This document contains lecture notes, classroom activities, homework problems, and challenge problems for Carroll College's MA342 - Numerical Analysis class. The content herein is written and maintained by Dr. Eric Sullivan of Carroll College. The textbook for the class is: Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms by Anne Greenbaum and Timothy Chartier, Princeton University Press, 2012. Reference to textbook problems are references to this version of Greenbaum and Chartier's book. Other problems were either created by Dr. Sullivan or are cited in the source code for the document (and are hence invisible to the student).

This material is written with an Inquiry-Based Learning (IBL) flavor. In that sense, this document could be used (almost) as a stand-alone set of materials for the course. The students are encouraged to work through problems and homework, present their findings, and work together when appropriate.

This class is roughly organized as follows:

  • Floating point arithmetic: Here we focus on how we store numbers in a computer.
  • Numerical algebra: Of particular interest in this chapter will be the numerical solution to algebraic equations. We will also have a look at Taylor's Theorem and see how it can help us quantify the amount of error that we're making with our approximations.
  • Numerical calculus: The focus of this chapter is to approximate derivatives and integrals in a single variable. We will find that Taylor's Theorem will play a role here as well as we build successively better and better approximations for the basic operators of calculus.
  • Numerical Linear Algebra: In this chapter we revisit much of the linear algebra that you already know (and some that you haven't seen), but we do so from the standpoint of learning efficient algorithms to do the computations. We will see several matrix factorizations for the first time. Exciting!
  • Numerical ODEs: We will revisit many of the numerical methods that you already know for numerical differential equations (e.g. Euler's and Runge-Kutta) but we will also introduce a few more. We will also spend time look at the stability of numerical methods for ODEs.
  • Numerical PDEs: In this final chapter we will examine the finite difference method for approximating solutions to partial differential equations. This is a huge area of mathematics so we are only going to talk about the primary elliptic, hyperbolic, and parabolic partial differential equations: the Laplace equations, the wave equation, and the heat equation. We will examine these equations in both 1 and 2 dimensions.