Instructor: Anil Damle
Office hours: Mondays and Wednesdays from 2:30 PM to 3:30 PM in 423 Gates Hall
TA: David Eriksson
Office hours: Tuesdays from 9:30 AM - 10:30 AM and Thursdays from 11 AM - 12 PM in 657 Frank H.T. Rhodes Hall, Room 2. (See David's website, linked to above, for directions.)
Lectures: Monday, Wednesday, and Friday from 11:15 am till 12:05 pm in Hollister Hall 306
Course overview: Stable and efficient algorithms for linear equations, least squares, and eigenvalue problems. Both direct and iterative methods are considered. Specific examples include QR and LU factorizations, Krylov subspace methods, the QR algorithm, and stationary iterative methods.
Your grade in this course will be determined based on your performance on the homeworks and exams. Please also read through the given references for each lecture.
There will be several homework assignments (tentatively six) throughout the course, typically made available roughly two weeks before the due date. They will include a mix of mathematical questions and implementation of algorithms. Implementation may be done in MATLAB, Julia, or Python. Homeworks must be typeset and submitted along with the associated code via the CMS.
There will be two take-home exams in this course, a midterm and a final. These exams are open book and note. However, you may not discuss the exam with anyone (besides asking myself or the TA clarifying questions) and are expected to complete them on your own. We reserve the right to use software systems (e.g., MOSS) to check for solution similarity. Implementation may be done in MATLAB, Julia, or Python. Your exam solutions must be typeset and submitted along with the associated code via the CMS.
Your grade on individual assignments will be determined based on both the correctness of your solutions and the clarity of their exposition (e.g., plots should be readable and clearly articulate what you are asked to show). Your final grade in the course will be computed based on the homework and exams in the following manner:
You are encouraged to actively participate in class. This can take the form of asking questions in class, responding to questions to the class, and actively asking/answering questions on the online discussion board. We will also be soliciting feedback mid-semester to hopefully improve the course.
You may discuss the homework freely with other students, but please refrain from looking at code or writeups by others. You must ultimately implement your own code and write up your own solution. In contrast, the take home exams are to be completed yourself, and should not be discussed with anyone (besides asking myself or the TA clarifying questions).
Except for the final exam (see above), all work is due at 11:59 pm on the due date. Homeworks and exams must be submitted via the CMS. For each homework assignment you are allowed up to two "slip days". You may not use slip days for the take-home exams.
Grades will be posted to the CMS, and regrade requests must be submitted within one week.
This course will require a mathematical background, particularly in linear algebra though we will also use some calculus and approximation theory. There will also be implementation based questions on the homeworks and exams, and therefore some programming knowledge is required as well.
The Cornell Code of Academic Integrity applies to this course.
In compliance with the Cornell University policy and equal access laws, I am available to discuss appropriate academic accommodations that may be required for student with disabilities. Requests for academic accommodations are to be made during the first three weeks of the semester, except for unusual circumstances, so arrangements can be made. Students are encouraged to register with Student Disability Services to verify their eligibility for appropriate accommodations.
We will not be explicitly following any single textbook in this course. Nevertheless, the books by Golub and Van Loan, and Trefethen and Bau collectively cover the material for the course and are recommended. The book by Demmel is also a good reference for a graduate course, and may prove valuable as an additional source.
In addition to the above textbooks there are numerous other online resources that you may find useful listed below. In particular, the first two provide coverage of much of the background material for this course.
A tentative schedule follows, and includes the topics we will be covering, relevant reference material, and assignment information. It is quite possible the specific topics covered on a given day will change slightly. This is particularly true for the lectures in the latter part of the course, and this schedule will be updated as necessary.
|8/31||Floating point||TB: 13, GVL: 2.7|
|9/3||Labor Day, no class|
|9/5||Sensitivity and conditioning||TB: 12, 14, and 15|
|9/7||Sensitivity and conditioning||TB: 12, 14, and 15||Homework 1 due|
|9/10||Sensitivity and conditioning||TB: 12, 14, and 15|
|9/12||LU factorizations||TB: 20-22 and GVL: 3.1-3.5|
|9/14||LU factorizations||TB: 20-22 and GVL: 3.1-3.5|
|9/17||LU factorizations||TB: 20-22 and GVL: 3.1-3.5|
|9/19||LU factorizations||TB: 20-22 and GVL: 3.1-3.5||Homework 2 due|
|9/21||Cholesky factorization||TB: 23 and GVL: 4.2|
|9/24||Backward stability and projectors||TB: 6 and 16|
|9/26||QR Factorizations||TB: 7-8 and GVL: 5.1-5.2|
|9/28||QR Factorizations||TB: 7-8 and GVL: 5.1-5.2|
|10/1||QR Factorizations||TB: 7-8 and GVL: 5.1-5.2|
|10/3||Least Squares Problems||TB: 11 and 18||Homework 3 due|
|10/5||Least Squares Problems||TB: 11 and 18|
|10/8||Fall break, no class|
|10/10||Instructor travel, no class||Midterm exam made available|
|10/12||Start iterative methods|
|10/15||Stationary iterative methods||GVL 11.2||Midterm exam due|
|10/17||Start Krylov subspace methods||TB: 32, 33, and 36; GVL: 10.1, 10.5, 11.3, and 11.4|
|10/19||Lanczos and Arnoldi||TB: 32, 33, and 36; GVL: 10.1, 10.5, 11.3, and 11.4|
|10/22||CG and MINRES||TB: 32, 33, and 36; GVL: 10.1, 10.5, 11.3, and 11.4|
|10/24||CG and MINRES||TB: 32, 33, and 36; GVL: 10.1, 10.5, 11.3, and 11.4|
|10/26||CG and MINRES||TB: 32, 33, and 36; GVL: 10.1, 10.5, 11.3, and 11.4|
|10/29||CG and MINRES||TB: 32, 33, and 36; GVL: 10.1, 10.5, 11.3, and 11.4|
|10/31||CG and MINRES||TB: 32, 33, and 36; GVL: 10.1, 10.5, 11.3, and 11.4||Homework 4 due|
|11/7||Eigenvalue Algorithms||TB: 24, 25, and 27|
|11/9||Power method, Rayleigh quotient iteration||TB: 24, 25, and 27; GVL: 7.3 and 8.2|
|11/12||Power method, Rayleigh quotient iteration||TB: 24, 25, and 27; GVL: 7.3 and 8.2|
|11/14||Orthogonal iteration||TB: 24, 25, and 27; GVL: 7.3 and 8.2|
|11/16||QR algorithm||TB: 28 and 29; GVL: 7.4, 7.5, and 8.3|
|11/19||QR algorithm||TB: 28 and 29; GVL: 7.4, 7.5, and 8.3||Homework 5 due|
|11/21||Thanksgiving break, no class|
|11/23||Thanksgiving break, no class|
|11/26||QR algorithm||TB: 28 and 29; GVL: 7.4, 7.5, and 8.3|
|11/28||Arnoldi and Lanczos for eigenvectors|
|11/30||Arnoldi and Lanczos for eigenvectors|
|12/3||Arnoldi and Lanczos for eigenvectors||Homework 6 due|
|12/12||Final exam||Final exam due at 11:30 AM|