MATH-UA 252 : Numerical Analysis
NYU Courant, Spring 2023
1. Logistics
Instructor | Samuel F. Potter (WWH 1104) |
Teaching Assistant | Mariana Martinez-Aguilar |
Lecture | Tu/Th 9:30-10:45 AM CIWW 201 |
Recitation | Fri 11:00-12:15PM 194M 307 |
(Note: first recitation is on 2/3) | |
Mariana's Office Hours | Wed 3:30-4:30 PM WWH 412 |
Sam's Office Hours | By appointment, or |
Thursdays after class |
1.1. Prerequisites
Calculus III and Linear Algebra. Programming is not officially a requirement for the class, but you can expect the course to be challenging if you have never programmed before. Likewise, if you have never seen or done a mathematical proof, you might have a turbulent time, even though the course is not as proof heavy as a class like real analysis.
1.2. Course Overview
After a short introduction, the first half of the class will essentially be an introduction to numerical linear algebra, while the second half of the class will cover classical topics in approximation theory and the solution of ODEs. In lectures, we will cover theory and algorithms; in the homework assignments, you will be exposed to more applications.
1.3. Logistics
There is a course BrightSpace. This is where you will submit your homework and receive your grades. I will also use BrightSpace to send announcements. It is your responsibility to read these announcements carefully. If you don't, you could easily overlook an important change in the course.
1.4. Attendance
Attendance is not mandatory. You do not need to ask for my permission to miss class. To prevent losing points because of a lack of attendance, you must present a valid excuse to me beforehand. This could involve a family emergency or a documented illness, in which case you would need a doctor's note.
1.5. COVID-19
It is your responsibility to be aware of NYU's current policies regarding COVID-19. Do not ask me to explain them to you. Do not send me pictures of COVID-19 rapid tests. Contracting COVID-19 is a valid reason for missing class. To demonstrate that you have COVID-19, you should send me confirmation of a positive PCR test or a doctor's note. Sending me a picture of a positive COVID-19 rapid test the night before an exam is not a valid reason for missing the exam. You must send me a positive PCR test or a doctor's note, which should be dated no later than one day after the exam in question.
1.6. Course materials
Unfortunately, there isn't a single textbook which covers the same topics in the same order as what I would like to present in class. I will provide references to sections from textbooks ahead of time which are you are free to read. It is not a requirement to do these readings, but obviously if you do them you will understand the material much better. These textbooks are also a good source of practice problems.
None of these textbooks are perfect. I've selected them primarily because they're available for free online through NYU's library. The textbooks I will use (along with links to where you can download them) are:
- Suli and Mayers, "An introduction to numerical analysis"
- Scott, "Numerical analysis"
- Griffiths and Higham, "Numerical methods for ordinary differential equations - initial value problems"
The material covered in this course is classical. Many books have been written about every topic you'll encounter during the semester. If these books don't do it for you, there's nothing stopping you from finding another book which you find a bit easier to understand. Some other books which you might find helpful but which are not available online are:
I will not provide references to these books, but if you choose to use them it shouldn't be difficult to match up what we cover in class with the relevant sections in these books.
1.7. Recitation
There is a weekly recitation, led by the TA. Attendance is not mandatory, but the recitation will be a good source of practice for the course. Materials related to the recitation may be posted online at the TA's discretion.
1.8. Homework
There will be 10 homeworks assigned throughout the semester. The first five will be due before the midterm, while the remaining five will be due after. The homeworks will include a mixture of written problems and programming exercises. Homeworks will be due midnight Monday the day before class. See BrightSpace for the exact dates. Homework solutions will not be posted.
1.9. Recitation worksheets
There will be 10 worksheets to be computed during recitation throughout the semester. Like the homework, there will be five during the first half of the course and five during the second half. Solutions to recitations will not be posted.
1.10. Exams
There will be a midterm and a final. For each exam, we will have a review session beforehand. Exams and exam solutions will not be posted. The final exam will be cumulative, and you will be allowed a one page (both sides) crib sheet for both exams.
1.11. Grading
Category | Percentage | Drops |
---|---|---|
Homework | 20% | 1st lowest |
Worksheets | 15% | 1st lowest |
Midterm | 30% | |
Final | 35% |
1.12. A few words about course logistics
Solutions to homeworks, worksheets, and exams will not be posted, because it is your responsibility to investigate your errors. Simply reading the correct solution will not be of much use. A better approach is to struggle to understand why your solution is incorrect. You are always welcome to come to office hours to ask for help if you don't know where to get started. Another option is to ask your friends. If enough people want help, I may go over some solutions in class.
2. Schedule
(Note: for the references "SM" refers to Suli and Mayers, "S" refers to Scott, and "GH" refers to Griffiths and Higham.)
Date | Week | Topic | References |
---|---|---|---|
1/24 | #1 | Solving nonlinear equations | S: Ch 1&2, SM: Ch 1 |
1/26 | Convergence of Newton's method | S: Ch 1&2, SM: Ch 1 | |
1/31 | #2 | Computer representation of numbers | Link, Link |
2/2 | Matrix norms, conditioning | SM: 2.7-8 | |
2/7 | #3 | The LU factorization | S: 3.1-3, SM: 2.3 |
2/9 | Pivoting | S: 3.4, SM: 2.4 | |
2/14 | #4 | Gram-Schmidt and QR | |
2/16 | Householder QR | ||
2/21 | #5 | Eigenvalue algorithms - Part I | |
2/23 | Eigenvalue algorithms - Part II | ||
2/28 | #6 | The SVD - Part I | |
3/2 | The SVD - Part II | ||
3/7 | #7 | Review | |
3/9 | Midterm #1 | ||
3/14 | Spring break - no class | ||
3/16 | Spring break - no class | ||
3/21 | #8 | Lagrange interpolation | |
3/23 | Hermite interpolation | ||
3/28 | #9 | The Vandermonde matrix | |
3/30 | The cardinal basis | ||
4/4 | #10 | Orthogonal polynomials | |
4/6 | Chebyshev polynomials | ||
4/11 | #11 | Newton-Cotes integration | |
4/13 | Gaussian quadrature | ||
4/18 | #12 | ODEs: Euler's method, Richardson extrapolation | |
4/20 | ODEs: local vs. global error, more methods | ||
4/25 | #13 | ODEs: consistency, stability, and convergence | |
4/27 | ODEs: A-stability | ||
5/2 | #14 | Trigonometric interpolation, the DFT | |
5/4 | Review |