Instructor: Anil Damle
Contact: damle@cornell.edu
Office hours: Monday 3 - 4 and Wednesday 2 - 3 in Gates 423
TA: Mariia Soroka
Contact: ms3663@cornell.edu
Office hours: Tuesday 5:30 - 6:30 in Rhodes 402 and Thursday 3:30 - 4:30 in Rhodes 404
TA: Yi Xu
Contact: yx622@cornell.edu
Office hours: Thursday 9 - 10 in Rhodes 400 and Friday 2 - 3 in Rhodes 404
Lectures: Monday, Wednesday, and Friday from 11:15 to 12:05 am in Gates Hall G01.
Course description: Introduction to the fundamentals of numerical linear algebra: direct and iterative methods for linear systems, eigenvalue problems, singular value decomposition. In the second half of the course, the above are used to build iterative methods for nonlinear systems and for multivariate optimization. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course.
Course websites: Homework and exams will be turned in using Gradescope and the course also has a discussion website using Ed.
Credit hours and grading schemes: This course is 4 credit hours and can be taken for a letter grade, S/U, or audit (auditing is only available to graduate students).
Prerequisites: This course requires sufficient mathematical maturity, particularly in linear algebra (e.g., MATH 2210 or MATH 2940) and multivariable calculus (e.g., MATH 2220 or MATH 1920), though we will also use some approximation theory. As we will also write proofs, it is also helpful to have add at least one additional 3000+ math course or relevant CS course (e.g., CS 2800 or CS 3220). There will also be implementation based questions on the homework and exams, and therefore some programming knowledge is required as well.
For undergraduates (i.e., those enrolled in CS 4220 or Math 4260), your grade in this course will be determined based on your performance on the homework, projects, and exams. For graduate students (i.e., those enrolled in CS 5223) there will be an additional paper review that consitutes part of your grade. 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. Homework solutions must be typeset and submitted along with the associated code via Gradescope.
There will be two open ended assignments, which we will refer to as projects. These assignments will present problems (with some guidance towards a solution) that you should be able to complete using the tools and techniques covered in the course. Implementation may be done in MATLAB, Python, or Julia. Projects (both code and a typeset report) will be submitted via Gradescope. Projects may be completed in groups of up to three students.
Graduate students will also have to complete a comprehensive paper review as part of this course. Your review must discuss what problem the paper is trying to solve or what question is being asked, the technical details of the solution/answer, how it fits in with related work, and a critique of the paper (e.g., did it satisfactorily address the key issues, were the numerical experiments convincing, are assumptions needed for theoretical claims reasonable, etc.). While your review is for a single paper, you may have to peruse some related papers to understand the area. Similarly, you could include short numerical experiments of your own, if applicable, as part of your critique.
The first deliverable will be around the midpoint of the semester and include the title of the paper you will review along with a brief justification for why you chose it. The paper can certainly be something related to your research, but it should be one you haven't gone through in full detail before. I have deliberately left the choice of paper wide open. It should be at least somewhat related to the topics of the course, and I am happy to provide suggestions if you are interested. Your review will be due at the end of the semester. Both the proposal and review must be typeset and will be submitted via Gradescope.
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 me 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 Gradescope.
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).
For CS 4220 and Math 4260 your final grade in the course will be computed based on the homework and exams in the following manner:
For CS 5223 your final grade in the course will be computed based on the homework and exams in the following manner:
The text by Ascher and Greif listed below serves as the primary reference text for the material covered in the class. Importantly, it is available for free online through Cornell's SIAM subscription. I have also included some additional references that may help prove to be useful resources.
In addition to the above textbooks there are numerous other online resources that you may find useful listed below. In particular, the first provides 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.
| Date | Topic | References | Notes/assignments |
|---|---|---|---|
| 1/22 | Introduction | The definition of numerical analysis by Nick Trefethen | |
| 1/24 | Linear algebra background | AG: 4 (and above references) | |
| 1/26 | Linear algebra background | AG: 4 (and above references) | |
| 1/29 | Numerical algorithms and error | AG: 1 | |
| 1/31 | Sensitivity and conditioning | AG: 1,2 | |
| 2/2 | Floating point | AG: 2 | |
| 2/5 | Floating point | AG: 2 | Homework 1 due |
| 2/7 | LU, pivoting | AG: 5.1, 5.2, and 5.3 | |
| 2/9 | LU, pivoting, Cholesky | AG: 5.1, 5.2, and 5.3 | |
| 2/12 | LU with complete pivoting and Choleksy | AG: 5.3 and 5.5 | |
| 2/14 | Sparse matrices | AG: 5.6 and 5.7 | |
| 2/16 | Least squares problems and QR factorizations | AG 6.1 and 6.2 | |
| 2/19 | QR factorization | AG 6.3 | Homework 2 due |
| 2/21 | QR factorization | AG 6.3 | |
| 2/23 | Ill-posedness and regularization | AG: 8.2 | |
| 2/26 | No class, February break | ||
| 2/28 | Intro to Eigenvalue problems | AG: 8.1 and Notes on perturbation theory | |
| 3/1 | Power and subspace iteration | AG: 8.1 | |
| 3/4 | Orthogonal iteration | AG: 8.3 | |
| 3/6 | QR algorithm | AG: 8.3 | |
| 3/8 | QR algorithm and the SVD | AG: 8.3 | Homework 3 due |
| 3/11 | Classical iterations | AG: 7.2 and 7.3 | Project 1 due |
| 3/13 | Krylov subspace methods | AG: 7.4 and 7.5 | Midterm posted |
| 3/15 | Krylov subspace methods: convergence | AG: 7.4 and 7.5 | |
| 3/18 | Krylov subspace methods: preconditioning | AG: 7.4 and 7.5 | |
| 3/20 | Nonlinear equations and optimization | AG: 9.1 | Midterm due |
| 3/22 | Intro to optimization | AG: 9.2 | |
| 3/25 | Search direction methods: gradient descent and newton | AG: 9.2 | |
| 3/27 | Line search | AG: 9.2 | Paper review proposal due 3/28 |
| 3/29 | Quasi-Newton Methods | AG: 9.2 | |
| 4/1 | No class, Spring break | ||
| 4/3 | No class, Spring break | ||
| 4/5 | No class, Spring break | ||
| 4/8 | BFGS | AG: 9.2 | |
| 4/10 | BFGS and practical tips | AG: 9.2 | |
| 4/12 | Gauss-Newton | AG: 9.2 | Homework 4 due |
| 4/15 | Iteratively reweighted least squares | AG: 9.2 | |
| 4/17 | Constraints | AG: 9.3 | |
| 4/19 | Constraints | AG: 9.3 | |
| 4/22 | Constraints | AG: 9.3 | |
| 4/24 | Constraints | ||
| 4/26 | Stochastic optimization | Section 5 of Optimization for Modern Data Analysis by Recht and Wright | Homework 5 due |
| 4/29 | Randomized NLA | SIAM Review paper by Halko, Martinsson, and Tropp | |
| 5/1 | Randomized NLA | SIAM Review paper by Halko, Martinsson, and Tropp | Project 2 due |
| 5/3 | Randomized NLA | SIAM Review paper by Halko, Martinsson, and Tropp | |
| 5/6 | Review | Homework 6 and paper review due (5/7) | |
| 5/16 | Final exam due May 16 at 12:00 pm |
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. Similarly, you may discuss the projects with other groups, but each group must ultimately produce their own code and report. In contrast, the take home exams are to be completed yourself, and should not be discussed with anyone. The paper review is also an individual assignment, though you are free to discuss the paper with others as long as you produce your own review.
Except for the final exam, all work is due at 11:59 pm on the due date. Homework, projects, and exams must be submitted via Gradescope. For each homework assignment, project, and the paper review you are allowed up to two "slip days". You may not use slip days for the take-home exams.
Grades will be posted to Gradescope, and regrade requests must be submitted within one week.
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.
You should expect and demand to be treated by your classmates and the course staff with respect. You belong here, and we are here to help you learn and enjoy this course. If any incident occurs that challenges this commitment to a supportive and inclusive environment, please let the instructors know so that the issue can be addressed. We are personally committed to this, and subscribe to the Computer Science Department’s Values of Inclusion. [Statement reproduced with permission from Dan Grossman.]
Cornell University provides a comprehensive set of mental health resources.