Numerical Linear Algebra

1.0 Title: Numerical Linear Algebra
Date: September 1, 2008
Credit Hours: 4
Prerequisite (s): 1016-331-Linear Algebra I (1016-432-Linear Algebra recommended)
Corequisite(s): None
Course proposed by: School of Mathematical Sciences

2.0 Course information:

Quarter(s) offered (check)
___ Fall __X__ Winter ____ Spring _____ Summer

Students required to take this course: (by program and year, as appropriate)
Applied and Computational Mathematics graduate students in the Scientific
Computing concentration.

Students who might elect to take the course:
Graduate students and advanced undergraduate students in mathematics,
physics, imaging science, and engineering.

3.0 Goals of the course (including rationale for the course, when appropriate):
To be able to apply matrix formulations in problem solving, to learn canonical
decompositions used in developing matrix-based algorithms and to be able to use
existing software packages in solving matrix -based problems

4.0 Course description (as it will appear in the RIT Catalog, including pre- and corequisites,
quarters offered)
1016–712 Numerical Linear Algebra
This course is a rigorous study of theoretical concepts and computational issues in
linear algebra . Topics include an analysis of Gaussian elimination with pivoting , its
error and its stability, iterative methods for solving linear systems, matrix
factorizations , eigenvalues, singular value decomposition , Krylov subspace methods
and application to least squares , systems of nonlinear equations and partial
differential equations . This course requires independent study of certain topics that
are not covered in the class lectures. Software packages like MATLAB will be
utilized through several computing projects. (1016-331, 1016-432 recommended)
Class 4, Credit 4 (W)

5.0 Possible resources (texts, references, computer packages, etc.)
5.1 Kincaid, D., Cheney, W., F., Numerical Analysis, Brooks/Cole
5.2 Stewart, G., Afternotes on Numerical Analysis, SIAM
5.3 Trefethen, L., Bau, D., Numerical Linear Algebra , SIAM
5.4 Tyrtyshnikov, E., A Brief Introduction to Numerical Analysis, Birkhauser

6.0 Topics (outline):

6.1 Direct methods for solving systems of linear equations
6.1.1. Gaussian elimination and back substitution
6.1.2. Pivoting
6.1.3. LU and Choleski decomposition

6.2 Error analysis
6.2.1. Vector and matrix norms
6.2.2. Rounding errors, forward and backward stability
6.2.3. Conditioning, perturbation analysis and residual

6.3 Iterative methods
6.3.1. Gauss-Jacobi and Gauss-Seidel
6.3.2. SOR

6.4 Eigenvalues
6.4.1. Power method , inverse method and shifts
6.4.2. Rayleigh quotient iteration
6.4.3. Orthogonal matrices and QR decomposition
6.4.4. QR algorithm
6.4.5. Schur decomposition
6.4.6. Hessenberg form
6.4.7. Singular value decomposition
6.4.8. Krylov subspace methods

6.5 Applications
6.5.1. Least squares , normal equations and pseudo -inverses
6.5.2. Newton and quasi-Newton methods for systems of nonlinear equations
6.5.3. Partial differential equations

7.0 Intended learning outcomes and associated assessment methods of those
outcomes

8.0 Program or general education goals supported by this course
8.1 To develop students’ understanding of the mathematical framework that supports
engineering, science, and mathematics.
8.2 To develop a capacity for critical and analytical thinking.
8.3 To develop an appropriate level of mathematical literacy and competency.

9.0 Other relevant information (such as special classroom, studio, or lab needs,
special scheduling, media requirements, etc.)
9.1 Computer laboratory facilities with appropriate software.

10.0 Supplemental information
None