CS 4220 / MATH 4260: Numerical Analysis: Linear and Nonlinear Problems

Instructor: Anil Damle

Contact: damle@cornell.edu

Office hours: Monday and Wednesday (starting January 29) from 10:00 AM - 11:00 AM in 423 Gates Hall, or by appointment.

TA: David Eriksson

Contact: dme65@cornell.edu

Office hours: Tuesday from 10:00 AM - 11:00 AM and Thursday from 1:00 PM - 2:00 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 2:30 pm till 3:20 pm in Olin Hall 165

Course overview: 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, projects, and exams will be turned in using the course management system (CMS). The course also has a discussion website using Piazza.

News and important dates

  • May 2 and 7 — Anil will not hold office hours due to travel
  • May 9 — Anil's office hours will be moved to 1:30 pm (till class)

Course work

Your grade in this course is comprised of three components: homework, projects, and exams. Please also read through the given references in concert with the lectures.

  • Homework:

    There will be five homework assignments 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 or Julia. Homeworks should be typed and submitted via the CMS.

  • Projects:

    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 or Julia. Projects (both code and a typed writeup) should be submitted via the CMS.

  • Exams:

    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. Implementation may be done in MATLAB or Julia. Your exam solutions and code should be typed and submitted via the CMS

    • Midterm exam (available via CMS)
    • Final exam (available via CMS at roughly 4:30 PM on Monday, May 7)


Your final grade in the course will be computed based on the homework, projects, exams, and course participation

  • Participation: 5%
  • Homework: 30%, 6% each
  • Projects: 20%, 10% each
  • Midterm exam: 20%
  • Final exam: 25%

Course policies


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.

Collaboration policy

You may discuss the homework and projects 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).

Late work and grading

Except for the final exam, all work is due at 11:59 pm on the due date. Homework and projects should be submitted via the CMS. For each assignment you are allowed up to two "slip days". However, over the course of the semester you may only use a total of six 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.


Linear algebra at the level of MATH 2210 or 2940 or equivalent and a CS 1 course in any language. We will assume you remember your calculus and can pick up MATLAB or Julia. Recommended but not essential: one additional mathematics course numbered 3000 or above. This course can be taken before or after CS 4210/MATH 4250.

Academic integrity

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.



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.

  • A First Course in Numerical Methods by Ascher and Greif
    This book serves as the primary reference for this course and covers a broad range of topics in numerical analysis. Sections are abbreviated as AG in the references below. [SIAM online]
  • Numerical Linear Algebra by Trefethen and Bau
    A good textbook that broadly covers numerical linear algebra, though at a graduate level. Nevertheless, it is well written, and may prove useful as a secondary source of information for some of the material. This book is also available online through the library. [Cornell library]
  • Applied Numerical Linear Algebra by Demmel
    A good textbook that covers numerical linear algebra, though at a graduate level and in significantly more technical detail than this course. Nevertheless, it may prove useful as a secondary source of information for some of the material, or as a vehicle for further study. This book is also available online through through Cornell's SIAM subscription. [SIAM online]


  • Lecture notes by Embree
    A set of lecture notes by Mark Embree for a course in numerical analysis he taught while at Rice. While the content of the notes does not completely align with this course, they still may be a useful and interesting resource. [online]
  • Linear algebra course by Strang
    Portions of this course will utilize your knowledge of linear algebra. If you feel you need additional preparation, or would like to revisit the topic, you may find Gilbert Strangs linear algebra course quite useful. [MIT Open Courseware]
  • Last years CS 4220 website
    Much of this course, at least topically, will match what was taught last year. Therefore, you may find the notes and other resources from last years class useful. [CS 4220 — Spring 2017]
  • Julia
    If you need resources for the Julia language, the documentation and various resources (under the learning link) are available on the Julia website. If consulting the documentation, please make sure the documentation you are consulting is for the version matching what you are running. [Julia webpage]
    One useful resource for MATLAB is the book "Insight through Computing: A MATLAB Introduction to Computational Science and Engineering" by Van Loan and Fan. It is available for free online through Cornell's SIAM subscription. [SIAM online] You may also find the online documentation helpful. [MATLAB documentation]


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
1/24 Introduction
1/26 Numerical algorithms and error AG: 1 (see 4 for the relevant linear algebra background)
1/29 Sensitivity and conditioning AG: 1,2
1/31 Floating point AG: 2
2/2 Linear algebra, background AG: 4
2/5 Linear Algebra, background AG: 4
2/7 Gaussian elimination and Cholesky AG: 5.1 Homework 1 due
2/9 LU and pivoting AG: 5.2 and 5.3
2/12 LU and pivoting AG: 5.2 and 5.3
2/14 Error analysis of direct methods AG: 5.4 and 5.8
2/16 Cholesky and Sparse matrices AG: 5.5, 5.6, and 5.7
2/21 Sparse matrices and intro to least squares problems AG: 5.6, 5.7, and 6.1
2/23 Least squares and QR AG: 6.2 and 6.3 Homework 2 due
2/26 QR factorizations AG: 6.2 and 6.3
2/28 QR factorizations via Householder AG: 6.2 and 6.3
3/2 Cornell closed, no class
3/5 Intro to eigenvalue problems AG: 8.1
3/7 Power and subspace iteration AG: 8.1 Project 1 due
3/9 QR iteration AG: 8.3 Midterm posted
3/12 Upper Hessenberg and tridiagonal forms AG: 8.3
3/14 Upper Hessenberg and tridiagonal forms
3/16 Optimization in 1D Midterm 1 due
3/19 Stationary iterative methods AG: 7.1, 7.2, and 7.3
3/21 Krylov subspaces and CG AG: 7.4 and 7.5
3/23 Preconditioning AG: 7.4 and 7.5
3/26 Nonlinear equations in 1D AG: 3.1, 3.2, and 3.4
3/28 Optimization in 1D
3/30 Nonlinear equations and optimization AG: 9.1 and 9.2 Homework 3 due
4/9 Newtons method AG: 9.1 and 9.2
4/11 Newton iterations AG: 9.1 and 9.2
4/13 Quasi-Newton and other iterations AG: 9.1 and 9.2
4/16 Gradient methods AG: 9.1 and 9.2
4/18 Line search AG: 9.1 and 9.2 Homework 4 due
4/20 BFGS AG: 9.1 and 9.2
4/23 BFGS and practical tips AG: 9.1 and 9.2
4/25 Modified Newton and trust region AG: 9.1 and 9.2
4/27 Constraints AG: 9.3 Project 2 due
4/30 Constraints AG: 9.3
5/2 Special topics
5/4 Review Homework 5 due
5/7 Review Final exam posted
5/9 Special topics
5/14 Final exam due at 4:30 PM Final exam due at 4:30 PM