UNC Course description
The Essentials
ID |
MATH566 |
Title |
Introduction to Numerical Analysis |
Times |
TR 2:00-3:15 PM, Peabody 220 |
Instructor |
Sorin Mitran mitran (AT) unc.edu |
Grading Assistant |
Anil Shenoy ashenoy (AT) email.unc.edu |
Office Hours |
Phillips 307, Mo: 12:00-1:00 PM, Tu: 12:30-1:45 PM, Th: 3:30-4:45 PM |
Description |
Basic numerical methods, their mathematical analysis and computer implementation |
Syllabus
- Iterative methods
- Interpolation: polynomial and spline approximations
- Numerical differentiation and integration
- Solution of ODEs and PDEs
Motivation and Objectives
Not all mathematical solutions can be expressed in closed, analytical form. Even when this is possible the analytical forms may be unwieldy. In such situations numerical methods offer an alternative that is widely used and the basis of the vast majority of current applications of mathematics in the sciences, engineering. This course introduces some basic methods in the context of real-life problems and presents the mathematical analysis of the methods as well as their implementation in Python.
Grading Policy
- Homework - 10 assignments x (3 1-point exercises + a 3-point computer application) = 60 points
- Midterm - 3 problems x 5 points = 15 points
- Final - 5 problems x 5 points = 25 points
- You will be given the opportunity during the Final Examination to answer a 5-point question from the midterm material (the question will be different from those given during the Midterm Examination). The score from this question will replace your worst score on a problem from the Midterm Examination.
Extra credit - Turn in any of these before Fall break to receive extra credit points. Note: email extra credit writeup (in pdf format) along with code you've written to mitran (AT) unc.edu. When e-mailing extra credit, use subject line: MATH566 (Last Name) ECx
EC1 (1 point) - Implement the bisection algorithm to solve f(x)=0 with a stopping criterion given by constant sign of the second derivative
EC2 (1 point) - Test how well EC1 works by applying Newton's method to the best approximation returned by bisection with
constant sign stopping criterion. Try a number of example f(x) (5 or more) and see if Newton's method reliably converges. EC3 (1 point) - Compare work done in bisection with
stopping criterion vs. bisection+Newton as in EC1+EC2. Which strategy results in fewer function evaluations? Again use 5 or more example f(x)'s
- Turn in any of these up to last day of class to receive extra credit.
- EC4 (3 points) - Repeat problem 4 from homework 7 using B-spline interpolation of degree 2,3,4 instead of piecewise quadratic interpolation
Mapping of point score to grades |
|||||||
|
|
B+ |
86-90 |
C+ |
71-75 |
D+ |
56-60 |
A |
96- |
B |
81-85 |
C |
66-70 |
D |
51-55 |
A- |
91-95 |
B- |
76-80 |
C- |
61-65 |
F |
-49 |
Course Texts and Notes
An Introduction to Numerical Methods and Analysis, James Epperson, Wiley, 2007.
Lesson Plan
Date |
Text |
References |
Date |
Text |
References |
8/25 |
1.1-1.2 |
8/27 |
1.3,1.5,1.6 |
||
9/1 |
1.4 |
9/3 |
2.1-2.3 |
||
9/8 |
2.4-2.5 |
9/10 |
2.6-2.7 |
|
|
9/15 |
3.1-3.3 |
9/17 |
3.4-3.7 |
||
9/22 |
3.8-3.9 |
|
9/24 |
3.10-3.11 |
|
9/29 |
4.1 |
|
10/1 |
4.2-4.4 |
|
10/6 |
Midterm review |
|
10/8 |
4.5-4.6 |
|
10/13 |
Midterm review |
|
10/15 |
Midterm Exam (1.1-4.6) |
|
10/20 |
Midterm discussion |
10/22 |
Fall Break |
||
10/27 |
4.7-4.8 |
|
10/29 |
4.8-4.9 |
|
11/3 |
4.10 |
|
11/5 |
5.1-5.4 |
|
11/10 |
5.6 |
|
11/12 |
5.7-5.8 |
|
11/17 |
6.1-6.3 |
|
11/19 |
6.4 |
|
11/24 |
6.5 |
|
11/26 |
Thanksgiving |
|
12/1 |
6.6 |
|
12/3 |
6.7 |
|
12/8 |
6.8 |
|
12/17 |
Final exam (4.10-6.11), 12:00 PM |
|
Homework
Homework is due on Tuesday, not accepted after Thursday, handed back the next Tuesday if turned in on time. It is preferred to present typeset answers to the 3 exercises, but handwritten solutions are accepted. The answer to the computer exercise must be typeset, the relevant code must be included along with tables and graphs presenting results.
Dates |
Exercises |
Computer work |
Solution |
Dates |
Exercises |
Computer work |
Solution |
9/1-9/8 |
p.12-13, Ex. 11,12,22 |
Ex. 15 |
9/8-9/15 |
p.50 Ex. 6,8, p. 72 Ex. 9 |
p. 73 Ex. 17 |
||
9/15-9/22 |
p. 92 Ex. 1,4 p. 99 Ex. 3 |
p.102-103 Ex. 4 & 5 |
9/22-9/29 |
p.124:5, p.154:5,9 |
p.155:13&14 |
||
9/29-10/6 |
p.166: 7 p.172-3:2,4 |
p.156: 23 |
10/6-10/13 |
|
|||
10/27-11/3 |
|
|
11/3-11/10 |
|
|
||
11/10-11/17 |
|
|
11/17-11/24 |
|
|
- Handwritten homework may be turned in at end of class or placed in the mailbox of Shenoy in the Mathematics Office Phillips 329
e-mail homework in pdf format to Anil Shenoy
When e-mailing homework, use subject line: MATH566 (Last Name) HWx
Hesabu - A Scientific Computing Software Environment
An immense amount of scientific computing software is freely available especially for Linux operating systems. As an introduction to what is available a virtual workstation environment has been prepared for this course. It is based on the Ubuntu distribution of the Linux operating system. Ubuntu is a Swahili word meaning 'Humanity to others'. The particular variant provided here is known as Hesabu which is the Swahili word for mathematics. There are 3 ways to access this environment:
- You can install Hesabu on your laptop
- You can run Hesabu in a Virtual Environment running inside Windows
- You can access the UNC Virtual Computing Lab
The main features of the Hesabu environment which we shall use are:
- Python with all scientific computation packages
TeXmacs and LaTeX
- Fortran, C/C++ compilers
- data visualization and plotting software
You might wish to use commercial packages for which UNC has campus licenses. You are welcome to do so, but most of the capabilities of these packages are available in the free software provided in Hesabu, often in a more efficient way. All examples in the course will use freely available software.
Examinations
Course Bulletin Boards
Please choose the appropriate section for your question. Questions are answered within 2 weekdays.