Math 449

Numerical Applied Mathematics, Fall 2024

Contact Information

Instructor: Ari Stern
Email: stern@wustl.edu
Office: Cupples I 211B
Office Hours: M 4:00-5:30pm, W 1:00-2:30pm

Catalog Description

Computer arithmetic, error propagation, condition number and stability; mathematical modeling, approximation and convergence; roots of functions; calculus of finite differences; implicit and explicit methods for initial and boundary value problems; numerical integration; numerical solution of linear systems, matrix equations, and eigensystems; Fourier transforms; optimization. Various software packages may be introduced and used. Prerequisites: Math 217 or 312, Math 309, Math 310 and CSE 131 (or other computer background with permission of the instructor).

Course Overview

This course is an introduction to numerical analysis, the branch of mathematics underlying scientific computing. For problems that cannot be solved by a closed-form formula (i.e., most problems), we can often use numerical algorithms to get an approximate answer to any desired level of accuracy. Numerical algorithms generally operate on real numbers (or their floating-point representations), in contrast to the core algorithms of computer science that involve integers and other discrete objects.

The numerical algorithms we will study include those for solving nonlinear equations and systems of equations, linear algebra, interpolation, and quadrature (i.e., computing integrals); numerical methods for differential equations will be postponed to Math 450 in the spring. Lectures will focus primarily on the mathematical theory of numerical analysis, while homework assignments will contain a mix of theoretical and computer programming problems.

Learning Objectives

1. Given a mathematical problem, identify an appropriate numerical algorithm to solve it.
2. Analyze the stability, convergence, and accuracy of various numerical algorithms (and understand what these properties mean).
3. Implement numerical algorithms as computer programs, and interpret the output of these programs in relation to the theoretical properties of the algorithms.

Lectures

Lectures will be held MWF 3:00-3:50pm in Seigle L006. The first class will be on Monday, August 26, and the last will be on Friday, December 8. Class will be canceled for Labor Day (Monday, September 2), Fall Break (Monday, October 7), and Thanksgiving Break (Wednesday, November 27, and Friday, November 29).

Homework Assignments

Problem sets will be posted to Canvas approximately biweekly, and students will submit their solutions electronically by uploading them to Gradescope before the specified due date and time. You are encouraged to discuss the homework with your fellow students, but your final write-up and all code must be your own. Please make sure that your solutions are written clearly and legibly. Late homework is generally not accepted.

Exams

There will be one in-class midterm exam on Wednesday, October 16. The final exam will be held on Thursday, December 12, from 6-8pm, location TBA.

Grading

Grades will be based on a weighted average of homework (40%, lowest score dropped), midterm exam (20%), and final exam (40%).

Pass/Fail policy: You must earn at least a letter grade of C- to get a P.

Required and Supplemental Texts

The required textbook for this course is An Introduction to Numerical Analysis, by Endre Süli and David Mayers (Cambridge University Press, 2003). This text is available as an eBook through the WashU Library, and a physical copy will also be placed on reserve.

As a supplemental text, I also recommend Computational Mathematics: An Introduction to Numerical Analysis and Scientific Computing with Python, by Dimitrios Mitsotakis (CRC Press, 2023), which contains additional material on programming and implementing numerical algorithms in Python. This is also available as an eBook through the WashU Library.

Software

The programming component of this class is based on the Python programming language, together with a collection of widely-used numerical and scientific computing libraries including NumPy, SciPy, and Matplotlib. No previous experience with Python or these libraries is assumed, although experience with some programming language is a prerequisite for this course. All of the software used in this course is free and open-source and can be installed on your own computer.

The Anaconda Python Distribution is officially recommended for this course, and is available for Linux, Mac, and Windows.

Academic Integrity

All students are expected to adhere to high standards of academic integrity, as specified in the undergraduate student academic integrity policy and graduate school academic and professional integrity policy. Any violations of this policy will be referred to the relevant Academic Integrity Officer. Violations of this policy include, but are not limited to:

• homework collaboration exceeding that permitted in the Homework Assignments section above (e.g., preparing your final write-up side-by-side with another student rather than independently, copying another student's solutions or code);
• submitting the work of others as your own (e.g., copying solutions or code found online);
• exam misconduct (e.g., copying another student's exam or knowingly allowing your exam to be copied, giving or receiving unauthorized assistance, use of unauthorized books/notes or other materials, unauthorized use of electronic devices, etc.).

If you have any questions, or are unsure about what is permitted/prohibited by this policy, please ask me.

In many cases, academic integrity violations are the result of getting behind in coursework and making bad decisions under pressure. Keep up with your assignments, ask questions when you are unsure what is expected of you, and do not give in to the temptation to cut corners.

COVID-19 Health and Safety Protocols

Students experiencing symptoms consistent with COVID-19 or concerned about a possible exposure should contact Habif Health and Wellness Center (314-935-6666) to arrange for testing as indicated. If a student tests positive for Covid-19, they will receive a letter with instructions about any necessary isolation that they can share with their instructors. Any accommodation needs for COVID-related absence not covered in an instructor’s standard course policies should be discussed between the student and instructor.

While on campus, it is imperative that students follow all public health guidelines established to reduce the risk of COVID-19 transmission within our community.

Access to Classroom Content During Excused Absences (Including COVID-19)

To encourage in-person attendance and participation, WashU no longer requires that lectures be recorded or otherwise made available outside the classroom, aside from certain exceptional cases: "In the case of excused student absences due to COVID and other factors (e.g., illness, religious holidays, family emergencies, etc.), instructors should develop strategies for providing students access to the fundamental content of a given class session so that students are able to make progress in the course while complying with public health and university guidelines around quarantine and isolation and managing other challenges that disrupt their ability to attend class."

If you cannot attend class due to COVID or other factors similar to those listed above, and if you contact me at least 3 hours prior to the start of class, I will try to arrange for video recording of the lecture. If I am unable to record the lecture (e.g., due to technical difficulties) or do not receive advance notice to do so, I will provide written lecture notes instead.

University Policies, Procedures, and Resources

Please visit this page for a list of general university policies, procedures, and resources.