MATH 348: Scientific Computation in Numerical Analysis

General Information

Instructor: Oscar Gonzalez (og@math.utexas.edu) 471-1138

Office hrs: RLM 10.122, Tu 11-12, W 2-3:30, Th 8-9

Lecture: RLM 6.120, MWF 10-11

Text 1: Numerical analysis, by Burden and Faires (9th Edition, Brooks/Cole, 2011)

Text 2: An introduction to C++ and numerical methods, by Ortega and Grimshaw (Oxford, 1999)

Course webpage: http://www.ma.utexas.edu/~og/TEACHING/M348/m348.html

Description

This course covers a variety of topics in the theory and application of numerical methods and provides practice in computer programming. The main topics include computer arithmetic, nonlinear algebraic equations, polynomial interpolation, numerical differentiation and integration, initial-value problems for ordinary differential equations, and direct methods for systems of linear algebraic equations. The objective is to prepare students for advanced studies in all areas of computational science and engineering. Emphasis will be placed on concepts and algorithms rather than proofs. Prerequisites: M408D; M341 or M340L; CS303E or CS307 or equivalent exposure to basic programming; all with a grade of at least C-.

Policies

Homeworks: Homework sets will be assigned weekly throughout the semester and will contain a mix of analytical and programming problems. It is acceptable for students to help each other with the homework sets; however, each student must write up and submit their own work. Both commented source code and sample output should be submitted with programming assignments. Computers can be found in the Undergraduate Computer Lab, RLM 7.122.

Programming: Computer programming will be a required part of the course, and the official language will be C++. No prior experience with C++ is necessary. Students will be introduced to the language through various assignments from the C++ text, sample programs, and online resources. The C++ text will be used for self-study; it will not be discussed in class.

Exams: There will be two midterm exams and a final exam. The tentative dates are

Exam 1 (Feb 25),

Exam 2 (Apr 8),

Exam 3 (May 13, 9am-12pm).

Exams 1,2 (midterms) will given in class. Exam 3 (final exam) will be given at the date, time and place scheduled by the University.

Course grade: Your course letter grade will be based on your homework and exam grades. The weights are

Homework average (35%),

Exam 1 (20%),

Exam 2 (20%),

Exam 3 (25%).

In the computation of the homework average, one homework grade (lowest) will be dropped.

Grade scale: Homeworks and exams will be assigned grades based on a 100-point scale. The weighted sum of your homework average and exam grades will determine your course letter grade as follows:

A (100-93),	A- (92-90),	B+ (89-87),	B (86-83),	B- (82-80),	C+ (79-77),	C (76-73),	C- (72-70),
		D+ (69-67),	D (66-63),	D- (62-60),	F (59-0).

Students with disabilities: The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.

Drop dates: The last day to drop the course without possible academic penalty is February 14, 2011. The last day to drop the course for academic reasons is March 28, 2011. After this, students may go to the Dean's Office to appeal for non-academic reasons.

Tentative schedule

The following is a tentative schedule for the core topics of the course. It does not reflect programming topics which students must cover independently. Please be aware that material may be reordered, added or deleted.
1. Computer arithmetic (1 week)

• Machine numbers
  
• Round-off errors
  
• Accumulation of errors
  

2. Equations in one variable (2 weeks)

• Bisection method
  
• Fixed-point iteration
  
• Newton and related methods
  

3. Interpolation and polynomial approximation (2 weeks)

• Lagrange polynomials
  
• Hermite polynomials
  
• Piecewise cubic interpolation
  

4. Numerical differentiation and integration (3 weeks)

• Multi-point differencing
  
• Richardson extrapolation
  
• Newton-Cotes quadrature
  
• Gauss quadrature
  
• Multi-dimensional problems
  

5. Initial-value problems for ordinary differential equations (3 weeks)

• Euler and Taylor methods
  
• Runge-Kutta methods
  
• Multi-step methods
  
• A-Stability and Stiffness
  

6. Direct methods for solving linear systems (3 weeks)

• Gaussian elimination and LU factorization
  
• Stability and condition number
  
• Pivoting strategies
  
• Choleski and Crout algorithms