# 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 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