Syllabus: MTH 651, Fall 2018

Advanced Numerical Analysis I

Instructor: Jay Gopalakrishnan
Times: Tue, Thu: 08:30-09:45
Venue: HAR 104 (Harder House)
Office hours: HAR 105, 10:00-11:00

Learning Outcomes

The course aims to teach often needed tools for numerical computations with linear operators (matrices), keeping in mind commonly arising sources of these operators (such as from discretization of partial differential equations). At the end of this course, a student would have gained facility with numerical algorithms, analytical techniques and software tools for working with large-scale linear systems.


Standard undergraduate linear algebra background is required for this graduate course.

Topical outline

Review of basic techniques for small dense matrices
Projections, SVD, least-squares, other factorizations
Sources of large sparse matrices, finite elements, finite differences
Conjugate gradients, Krylov spaces
Arnoldi, Lanczos, GMRES & other Krylov space techniques
Introduction to preconditioners
Algorithmic approach to multigrid
Eigensolvers, convergence analysis using gap

For a more detailed daily schedule, please see the class diary (after logging to D2L).

Learning methods

Theory: Most of the mathematical theory will be covered in lectures. Exercises will be given throughout. Some of the meeting times will be devoted to solving exercise problems.

Code: In this course, we shall use Python 3 and its SciPy and NumPy modules (none of which requires any purchase for downloading and installation). Some pointers for those who are not already familiar with these:

  • For Python, a good starting point is it's inventor's tutorial.
  • For Scipy, a good starting point is Scipy lectures.

To learn to implement algorithms, some meeting times will reserved for hands-on software sessions. Students are expected to bring their own laptops for these sessions.

For an algorithmic introduction to finite elements, we will use the open source software NGSolve. Help with installation and introduction to its python interface will be offered during the course. During this term, we shall acquire facility with manipulating the matrices generated by finite elements. We shall continue to use this software in the sequel during the next term, where the theory of finite elements is covered in depth.

Textbook: Students are not required to buy any textbook for this course. The learning material will not come from a single source. These reference books will be helpful:

  • Numerical Linear Algebra, by Lloyd N. Trefethen and David Bau III.
  • Matrix Computations, by Gene H. Golub and Charles F. van Loan
  • Iterative Methods for Sparse Linear Systems by Y. Saad

Course Management System: We will use the university's D2L, where (after logging in) this course's landing page can be found. All course materials will be placed there.

Evaluation of learning

Grades will be assigned based on take-home exams/projects.

Fine print

Academic Misconduct: In the list of prohibited student behavior at PSU is plagiarism, buying and selling of course assignments, and obstruction of another student's success. Students are expected to know of and refrain from all proscribed conduct.
Title IX Reporting Obligations: Every instructor at PSU has the responsibility to help create a safe learning environment for students and for the campus as a whole. As a member of the university community, an instructor must report any instances of sexual harassment, sexual violence and/or other forms of prohibited discrimination. If you would rather share information about sexual harassment, sexual violence or discrimination to a confidential employee who does not have this reporting responsibility, please use the online list of those individuals. For more information about Title IX please complete the student module Creating a Safe Campus in D2L.
Disability Accommodations: The Disability Resource Center (DRC) provides reasonable accommodations for students who encounter barriers in the learning environment. If you have, or think you may have, a disability that may affect your work in this class and feel you need accommodations, contact the DRC to schedule an appointment and initiate a conversation about reasonable accommodations. The DRC is located in 116 Smith Memorial Student Union, 503-725-4150,, Students who have testing accommodations must begin the test at the same time as the rest of the class.
Course Materials: All course materials handed out in class or placed in D2L are solely for the use of each student registered in this course. Sale of these materials is prohibited. During class sessions, voice or video recording of the instructor or other students without their explicit written consent is prohibited.

Author: Jay Gopalakrishnan

Last updated: 2018-09-29 Sat 11:17