UNIVERSITY OF TENNESSEE

SYLLABUS

Math 571 - Numerical Mathematics I

3 Credit Hours

Section 001, CRN 43243

Meetings: TR 12:40p--1:55p Ayres Hall 121.

Fall 2017

Course Description: Direct and iterative methods for linear systems. The algebraic eigenvalue problem and the singular decomposition theorem. Newton and quasi-Newton methods for systems of nonlinear equations.

Office Hours and Location: Wednesdays 2--3pm in Ayres 204. Office hours can also be arranged by appointment.

Course Communications: You may write to me at asalgad1@utk.edu with questions, comments, etc. To avoid confusion, please write the course name (MATH571) in the Subject line. You should use your university e-mail account when sending me emails.

Course web page: Here

Goals: The successful student will be able to derive, apply, and analyze elementary numerical algorithms. This course (together with Math 572) is intended to prepare mathematics students for the numerical mathematics preliminary examination. Topics in this class include numerical linear algebra, practical matrix factorizations, stability, least squares problems, root finding, iterative methods for large linear systems, and eigenvalue problems.

A good portion of this course will be concerned with the standard theorems of numerical linear and nonlinear algebra. To prove these results we will employ basic tools from linear algebra and differential and integral calculus, including matrix factorizations (LU, SVD, QR, etc.), the mean value theorem, Taylor's theorem, the intermediate value theorem, etc. Familiarity with these tools will be largely assumed.

Textbook:
Required:

• Numerical Linear Algebra, L.N. Trefethen and D. Bau, SIAM, 1997.

Supplementary:

• Matrix Computations, G.H. Golub and C.F. van Loan.
• Introduction to Numerical Analysis, J. Stoer and R. Bulirsch, Springer, 3rd edition, 2002.
• A Theoretical Introduction to Numerical Analysis. V.S. Ryaben'kii and S.V. Tsynkov, Chapman & Hall/CRC, 2007.

Course Requirements, Assessment and Evaluations:

• Homeworks will be handed out every other week, or can be downloaded from the class website. While generally homework will not be collected, doing homework will help with the quizzes and exams. In this class it is OK to try to solve the homework in groups. Quizzes and programming assignments, however, must be turn in on an individual basis.

• There will be two (2) programming assignments throughout the semester. You can solve them using Matlab©, or any programming language of your choice. Submission of these can be done online.

• There will be a short (10-15 minutes) quiz approximately every other week. They will generally cover the material of the previous lectures.

• There will be one midterm and a comprehensive final exam. A tentative date for the midterm is October 19. The official date will be announced at least two weeks in advance. The University Calendar sets the final examination date to be Tuesday, December 12 10:15am--12:15pm.

• The midterm is worth 30 points, the final is worth 35 points, quizzes are worth 20 points and programming assignments 15 points. The cutoffs will be approximately as follows: 90% or higher is an A, 80% - B, 70% - C and 60% - D.

Makeup Policy: Late programming assignments will NOT be accepted. Makeups for the quizzes, midterm and final will be given only if a student can present evidence that an absence was caused by serious illness, a death in the immediate family, religious observance, or participation in University activities at the request of University authorities. For an illness, you must present a signed statement from a doctor that your illness was sufficiently serious to make you miss class. A note saying only that you visited the doctor or the Health Center will not suffice.

Course Outline: This is a tentative list of topics to be covered in class.

1. Review of linear algebra.
2. The singular value decomposition theorem.
3. Overdetermined linear systems and least squares methods.
4. QR factorization.
5. Householder method for QR factorization.
6. Systems of linear equations and conditioning.
7. Gaussian elimination and its variants for simple systems.
8. LU factorization and pivoting strategies.
9. Cholesky factorization.
10. Iterative methods for the solution of linear systems.
12. Computing eigenvalues, Gershgorin theorems.
13. The QR algorithm for the computation of eigenvalues.
14. The inverse iteration and power method for the computation of eigenvalues. The Raleigh quotient.
15. Solution of nonlinear systems. Bisection, chord and secant methods.
16. Solution of nonlinear systems. Newton methods.

If the instructor finds it necessary to make informational changes (e.g.  office hours, schedule adjustments) due to students' needs or unforeseen circumstances, students will be notified in writing/email of any such changes.