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MATH 441 Syllabus

Introduction to Numerical Analysis

Revised: September 2020

Course Description

This first semester introduction to the field of numerical analysis will investigate numerical techniques in: solving equations in one variable, interpolation and polynomial approximation, numerical differentiation and integration, and solving ordinary differential equations.  The errors associated with these techniques will also be examined. 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, implement, apply, geometrically represent, and compare-contrast a variety of numerical algorithms in the fields of root-finding, interpolation, differentiation, integration, and solving initial value problems for ODEs;
  • Explain the derivation of error terms associated with any of these methods as well as the significance of the error term for use in an application;
  • 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 new but relevant problems.

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) [5 days]
  • Taylor’s Theorem
  • Computer Representation of Numbers
  • Floating Point Arithmetic
  1. Solutions of Equations in One Variable (Chapter 2) [10 days]
  • Bisection Method
  • Fixed-Point Iteration
  • Newton's Method
  • Variants of Newton’s Method
  • Müller's Method
  1. Polynomial Interpolation (Chapter 3) [6 days]
  • Lagrange Polynomial
  • Divided Differences
  • Cubic Spline Interpolation
  1. Numerical Differentiation and Integration (Chapter 4) [10 days]
  • Numerical Differentiation
  • Simple Quadrature Methods
  • Composite Methods
  • Adaptive Quadrature Methods
  1. 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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