MATH 441 Syllabus

Introduction to Numerical Analysis

Revised: September 2014

Course Description

This first semester introduction to the field of numerical analysis will investigate numerical techniques in: solving equations in one variable (a.k.a. root finding) interpolation and polynomial approximation, numerical differentiation and integration, and olving ordinary differential equations and the errors associated with each of these techniques. A significant component of the class comes from implementing or using these methods to complete projects. Prequisites: MATH 255, CS 150 or MATH 340. Three semester hours.

Student Learning Objectives

By the end of the course, students will be able to:

  • Explain the significance of floating point arithmetic and a computer's representation of floating point numbers to the accuracy of numerical computations as well as provide examples;
  • Explain the difference between, and calculate, absolute error and relative error for an approximation technique;
  • Explain, implement, apply, geometrically represent, compare and contrast the following:
    • Root-finding methods such as Bisection Method, Fixed Point Method, Secant Method, Newton's Method, Method of False Position, and Muller's Method;
    • Interpolation methods such as the nth order Lagrange polynomial, piecewise polynomials, and cubic splines;
    • Numerical differentiation methods such as Forward Difference, Backward Difference, and Centered Difference;
    • Numerical integration (quadrature) methods such as Midpoint Method, Simpson's Method, Trapezoid Method, Open Newton-Cotes Methods, Closed Newton-Cotes Methods -- either as single interval or composite methods -- and Adaptive Quadrature Methods;
    • Numerical initial value problem methods such as Euler's Method, Higher Order Taylor Methods, Runge-Kutta Methods, and Adaptive Runge-Kutta Methods;
    • Explain the derivation of error terms associated with any of the above methods as well as the significance of the error term for use in an application;
    • Interpret the purpose and actions of an algorithm given code for an unspecified numerical method;
    • Recognize the type of applied problem at hand and justify which numerical method is most appropriate for solving a particular instance of the problem; and
    • Extend methods discussed in class to apply them to a new but relevant problem.

Required Text

Burden & Faires, Numerical Analysis (9th Ed.), Thompson Brooks/Cole Publishing.

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

1. Preliminaries (Chapter 1) [4 days]

  • Round-off Errors

  • Floating Point Arithmetic

2. Solutions of Equations in One Variable (Chapter 2) [10 days]

  • Bisection Method

  • Fixed-Point Iteration

  • Newton's Method

  • Müller's Method

3. Polynomial Interpolation (Chapter 3) [6 days]

  • Lagrange Polynomial

  • Divided Differences

  • Cubic Spline Interpolation

4. Numerical Differentiation and Integration (Chapter 4) [11 days]

  • Numerical Differentiation

  • Simple Quadrature Methods

  • Composite Methods

  • Adaptive Quadrature Methods

5. Initial Value Problems for Ordinary Differential Equations (Chapter 5) [14 days]

  • Euler's Method

  • Higher-Order Taylor Methods

  • Runge-Kutta Methods

  • Multistep Methods

  • Higher Order Equations and Systems
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