Syllabus
| Catalog # | Math 280W |
| Instructor | Ophelia Adams |
| Lecture | Hylan 202 TR 9:40 – 10:55 |
| Office Hours | Hylan 813 TR 11:00 – 12:00 |
| ophelia.adams@rochester.edu |
Course Description
This course develops the mathematical machinery to describe and justify numerical techniques for solving problems in the natural sciences, such as inverting matrices, calculating eigenvectors/values, finding roots of equations, integrating functions, and solving differential equations; qualitative behavior of solutions both exact and numerical. Many of these problems cannot solved by analytic methods, or practical considerations make analytic methods and solutions less useful.
Beyond that, we will look at analyzing and translating genuine problems from other fields (engineering, chemistry, physics, &c) into mathematics with special attention to what is changed by this translation: how well does the mathematics capture the original question? what is lost, and what is preserved? what is the “physical” relevance (or meaning) of the methods developed? when and how does this justify the extra-mathematical value of our work?
Prerequisites:
- Linear algebra and proof writing: Math 235 or 173, or both Math 165 and 200 with instructor approval.
- Programming: CSC 161 or 171. Other programming background with instructor approval. We will use SageMath, a large extension of Python, so numpy and other familiar tools are available too.
Course Materials
- Course notes. Prepared by and with the help of Yuankun (Kunko) Zou and John Nguyen. Funded by my Teaching Center SCDP grant and my University Library OER grant, along with some startup funds from the mathematics department. (many thanks to all involved!!)
- CoCalc license. For accessing most course materials and working on projects. Software will be available through this, especially in the convenient Jupyter Notebook format, so that you do not need to install anything on your personal devices.
- Textbook. Numerical Analysis: Mathematics of Scientific Computing, 3rd Edition by David Kincaid and Ward Cheney.
- (optional) Various readings – books, chapters, articles – posted on Blackboard.
Inclusion Statement
I am committed to cultivating an inclusive and encouraging classroom environment where everyone is welcomed and supported to learn mathematics and achieve their personal learning goals. To help foster such an environment, please:
- let me know if your name or pronouns differ from those in the official roster (i.e. UR Student).
- let me know if there are issues outside of class which are affecting your performance in the course; I will try to be flexible and work with you to figure out a path for you to achieve success in this course.
- show respect and due consideration for your classmates’ views and ideas during discussions and all other peer interactions; group-work and collaboration are an important aspect of mathematics.
Course Objectives & Learning Outcomes
Numerical analysis is a multifaceted subject, drawing on tools from calculus, linear algebra, and more. We will aim to explore it thoroughly from several angles. Here are our primary course and learning objectives:
- (Re)acquaint ourselves with calculus in several variables, with special attention to the powerful tools it proves for approximation.
- (Re)acquaint ourselves with linear algebra, both in its own right and as an important tool in calculus and differential equations.
- (Re)acquaint ourselves with differential equations
- Refine our understanding of those prerequisites and the tools they provide already.
- Discuss new concepts and techniques for approximating solutions to linear, nonlinear, and differential equations.
- Analyze the numerical features of the methods themselves, such as speed and error.
- Implement these methods on a computer: both for the conceptual-learning value of rendering concrete seemingly-abstract mathematical principles, and to learn and practice the valuable programming skills expected by many careers in applied mathematics (and many other scientific and technical fields!).
- Study asymptotic and qualitative behavior, solutions and their quality, well-posedness.
- Assess problems arising in physics, chemistry, engineering, and more: describe their qualitative behavior, and determine likeliness of amenability to solution by our methods.
- Practice written and oral scientific communication skills.
Grading
There are three main components to how your learning is evaluated in this class: homework, writing (primarily projects), and exams. There will be ten or so homeworks, four journals, one large project, one midterm, and one final exam. These contribute to your grade as follows:
| Course component | % of grade | Notes |
|---|---|---|
| Homework | 20% | 15% Written, 5% Programming |
| Writing | 20% | 16% Project, 4% Journals |
| Midterm | 30% | |
| Final | 30% |
Homework
Homework is crucial to success in the course. One of the best ways to learn mathematics is to do mathematics. It is essentially impossible to master the material without engaging with it directly.
Homework is always due Friday at 10pm. We will accept homework up to a day late for a 10% penalty, or two days late for a 20% penalty. No assignments will be accepted after that, so that we can post the solutions promptly for the whole class. Each assignment is worth an equal portion of the homework portion of your grade. Homework is graded for correctness and clarity; partial credit is always available for good progress toward a solution. Programming assignments will, as appropriate, be tested against sample data for accuracy. Writing clear code is a valuable skill, but not one we will directly assess. However, it is easier for the grader to give partial credit for incorrect implementations if the comments make code more readable, or help them see what you were thinking.
You may collaborate with peers on these assignments, and we encourage you to do so, but you must (1) write up your solutions on your own, and (2) include at the top of your assignment the names of those with whom you worked; proper attribution is important.
You are always welcome and encouraged to come to office hours with questions!!
Writing Component
The written component of the course consists of several journaling assignments, one mini project, and one final project. The projects all have a significant programming component. This and written communication are extremely important skills for any career in mathematics and nearby fields.
This course satisfies the Type A upper level writing requirement for the mathematics majors. For administrative reasons, you must register for the separate 0.5 credit writing module, Math 280WM-1 to have it counted as such.
Exams
The midterm covers Units 0, 1, and 2. The final exam is cumulative, with special focus on Units 3 and 4. See calendar for more specific topics and readings.
- Midterm: Thursday October 9th, 9:40am – 10:55am, location TBD.
- Final: Monday December 15th, 4:00pm – 7:00pm, location TBD.
No notes, calculators, cell phones, or other electronics are allowed on the exams. Any uses of these or other material is a violation of the academic honesty policy.
Exams are largely proof-based and graded for correctness and clarity. Programming is not tested.
Tentative Calendar
The schedule is subject to some change, but we will always aim to provide as much notice as possible. The exams will not be rescheduled under any circumstances, nor will the units move; changes are limited to rearranging or removing topics within a unit, and possibly adjusting homework to match.
See the course website for the most up-to-date calendar information, including assignments.
Week
Topics
Comments
| 1 | Calculus, error propagation | |
| 2 | Linear algebra, some diffeq |
| 3 | Norms and condition number | |
| 4 | Markov matrices and iterative methods Generalized Jacobi method |
| 5 | Polynomial approximation | |
| 6 | Spline approximation | |
| 7 | Review and Midterm |
| 8 | Bisection method | fall break |
| 9 | Newton and secant methods | |
| 10 | Homotopy continuation |
| 11 | Numerical integration | |
| 12 | Taylor methods | |
| 13 | Runge-Kutta methods |
| 14 | Catch-up, projects | Thanksgiving |
| 15 | Wrap-up and review |
Policies
Disability Support
The University of Rochester respects and welcomes students of all backgrounds and abilities. In the event you encounter any barrier(s) to full participation in this course due to the impact of a disability, please contact the Office of Disability Resources. The access coordinators in the Office of Disability Resources can meet with you to discuss the barriers you are experiencing and explain the eligibility process for establishing academic accommodations. You can reach the Office of Disability Resources by email at disability@rochester.edu; by phone at (585) 276-5075; in person at Taylor Hall; online at their website.
Please note that to be granted alternate testing accommodations, you (the student) must fill out forms with Disability Resources one to two weeks in advance; these forms are not sent automatically. Instructors are not responsible for requesting your alternative testing accommodations or for providing accommodations on their own.
Academic Integrity Statement
All assignments and activities associated with this course must be performed in accordance with the University of Rochester’s Academic Honesty Policy.
You should only submit your own, original work. If discussion and collaboration are allowed (homework; not exams) you must write up your assignment individually and note who you worked with at the top for proper attribution. Suspicious work will be referred to the Board on Academic Honesty for review.
AI Policy
Use of AI tools (ChatGPT, Claude, co-pilot) is not allowed in any form on any assignments for this course. Suspected uses will be referred to the academic honesty board. Moreover, you should not use AI tools to write emails to the instructor, TAs, or other people affiliated with the course.
Math Department policy on unauthorized online resources:
Any usage whatsoever of online solution sets or paid online resources (chegg.com or similar) is considered an academic honesty violation and will be reported to the Board on Academic Honesty. If any assignment is found to contain content which originated from such sources, then it is typically subject to a minimum penalty of zero on the assignment and a full letter grade reduction at the end of the semester (e.g. a B would be reduced to a C). Depending on the circumstances, this may apply even if the unauthorized content was obtained through indirect means (through a friend for instance) and/or the student is seemingly unaware that the content originated from such sources. If you have any questions about whether resources are acceptable, please check with your instructor.
Credit Hour Policy
Math 280W is a 4.5 credit hour course, which follows the college credit hour policy for such courses. The main lectures meet for 3 academic hours per week. Per the credit hour policy, you plan to spend at least 10 further hours per week engaging with the material (reading the notes and book, reviewing your notes, completing homework, preparing your projects, and so on).
Student Conduct
As students at the University of Rochester, you have agreed to uphold the Standards of Student Conduct. See, in particular, the policy on harassment and discrimination. Conduct issues will be referred to the appropriate offices within the university as described in the linked document.