# MATH 620 Introduction to Numerical Analysis

#### Basic Information

• Dr. Manil Suri, Math/Psych 419, (410) 455-2311, suri@umbc.edu,
office hours: MW 3:30-4:30 or by appointment
• Lectures: MW 5:30-6:45, Meyerhoff Chemistry 256
• Text: An Introduction to Numerical Analysis by K. Atkinson (2nd edition)
• Syllabus: Chapters 1 (briefly), 2 (2.1-2.3, 2.5-2.6, 2.10-2.11), 3 (3.1-3.2, 3.5, 3.6-3.8 briefly), 4 (4.1, 4.3-4.5), 5 (5.1-5.3, 5.7), 6 (6.1-6.5, 6.8). Sections 7.3, 8.1-8.3 from Chapters 7 and 8 will be also covered briefly.
• Supplementary Reference Text: Numerical Analysis: An introduction, by Walter Gautschi. (For reference only, if needed)
• Prerequisites: MATH 221, MATH 301, familiarity with ODEs. You will need to know a programming language (such as MATLAB) to be able to write your own code.

#### Overview and Goals

Problems encountered in mathematics courses such as Calculus generally can be solved by paper and pencil, and yield nice, “closed form” answers in the form of numerical values, functions or formulas. In real world applications, however, problems are seldom so well behaved. For instance, integrals may be difficult or impossible to compute exactly, differential equations may have solutions that can only be expressed as infinite series, and systems of nonlinear equations may not have any solutions that can be found by hand. Consequently, a large proportion of problems that mathematicians, engineers, scientists and other professionals “solve” are only done numerically, using computer power. Math 620 introduces you to various types of such computational methods that can be used to tackle an array of mathematical problems (several of which you will have encountered before in other courses). It therefore provides an introduction to the field of “scientific computing.”

A common characteristic of the methods mentioned above is that they are based on approximations of some sort: a numerical value may be replaced by a decimal expansion close to it, a function by a polynomial expansion, a problem by a “nearby” problem that is easier to solve. One of the key issues that we will learn in this course is how to analyze and control the resulting error in our computed answers. Our primary emphasis will be to gain an understanding of these methods through theoretical analysis.

We will also perform computer experiments in our quest to gain familiarity with these methods. Theoretical results are often “asymptotic” in nature – our goal will be to see whether these results are observed in practice. It is only through performing experiments that one can develop a “computational sixth sense” to decide when computer solutions are to be trusted and when not.

Although you are free to use other languages to write programs for assignments, you should at least have basic familiarity with MATLAB.

The course will also prepare you for the Math 620 graduate comprehensive exam.

To summarize, the main goals of the course are:

• Learning computational methods that will solve problems (integration, differential equations, non-linear equations, etc) approximately
• Understanding, analyzing and assessing the errors in approximate solutions obtained through these methods
• Gaining experience in performing numerical computations
• Writing and using your own computational programs.
• Preparation for Math 620 graduate comprehensive exam.

#### Tests and Homework

• HOMEWORK is an essential part of the course. It will consist of both computer and paper and pencil problems. There will also be a project. Homework assignments will be posted on Blackboard and will be due every week or two.
• TEST This will be given around the middle of the semester. The date will be announced at least 2 weeks in advance.
• FINAL This will be cumulative, and will consist of a take-home part and an in-class part. The in-class final will be practice for the Math 620 comprehensive. It will be held in our regular classroom, on TBA, from 6 to 8 pm.
• MAKE-UPS will only be allowed under special circumstances with written documentation and prior approval if possible. If you miss something, contact me immediately (i.e. on that day) via e-mail (or phone).