# MATH 56: Computational Methods

Course Information:
• Instructor: Professor Anne Gelb, Mathematics Department, Dartmouth College
• Course Time: 10A T-Th 10:10am-12:00pm (x-hour F 3:30pm-4:20pm)
• Course Location: Kemeny 201
• Office: Kemeny 207 Office Hours: M 2:30-3:30 Th 1:00-2:00; and by appointment.
• Jonathan Lindbloom (TA) Office: Kemeny 219 Office Hours: TBD.

ORC Course Description: This course introduces computational algorithms solving problems from a variety of scientific disciplines. Mathematical models describing a phenomenon of interest are typically too complex to construct analytical solutions, leading us to numerical methods. Motivated by models from physics, biology, and medicine, students will develop numerical algorithms and mathematically analyze their accuracy, efficiency, and convergence properties. The course will provide external coding resources as students will implement algorithms in MATLAB. Sample topics include matrix decompositions, inverse problems, optimization, data fitting, and differential equations.

Course Goals: This course emphasizes the numerical analysis of classical computational methods, and is a fundamental topic in applied mathematics. An appropriate computational algorithm must maintain the physical properties of the underlying system matical properties it is supposed to solve. Moreover, the method must be stable to small perturbations inevitably introduced either as a consequence of reduced order modeling or noisy data collection, so that the results are deemed trustworthy by the domain scientists. Numerical analysis provides the methodology needed analyze a given computational method used for a particular class of problems, as well as the tools for developing new computational methods for modern problems. Finally, numerical analysts answer the question, ``what assumptions of the underlying problem are necessary for this computational method to succeed?'' Such a question is critical as we incorporate increasing amounts of information (coming both from experts and data collections).

In this course we will focus on numerical linear algebra, interpolation and approximation, which are all essential when solving problems in data science, signal and image processing, and evolutionary dynamics. We will use MATLAB to verify our understanding of the theoretical results, but developing programming skills is not the main focus of the course.

Prerequisites: Math 22 or instructor approval. Some experience in MATLAB or another programming language is expected.

Textbooks:

• Lambers, James V., Sumner Mooney, Amber C., and Montiforte, Vivian (2021), Explorations in Numerical Analysis: Python Edition, World Scientific Press (required). 2016 preprint of book (in MATLAB).
• Ascher, Uri M. and Greif, Chen. (2011) A First Course in Numerical Methods, SIAM (suggested).
The SIAM book is available as an e-book through the Dartmouth library.

Grading: Grades in the class will be based on homework sets which will ensure mastery of theoretical and computational skills. Students may work together on the homework, but will need to turn in their own assignments. Students may not work together on final take home exam, but may do a joint final project. It is strongly recommended that all homework assignments, especially those involving programming problems, be started early.

Grading formula: (i) Homework sets (70%); (ii) final project or take home exam (20%); (iii) Participation & Attendance (10%).

## Important dates and grading information

• First day of class: January 4 2023.
• 5 homework problem sets due approximately every ten days. Homework sets will be available on CANVAS. Due to the varying complexity of the material, some homework sets will naturally be more challenging than others. Regardless, each homework set is weighted the same for the final grade.
• X Hours will be used only as needed and will be held remotely.
• Participation & attendance: Students are expected to attend most classes and X hours (when scheduled).
• Last day of class: March 7 2023
• Final exam/project due: TBD

## Syllabus

### Tentative lecture plan which may be subject to further changes.

Week Lecture
Weeks 1 & 2 Chapters 1 & 2: Preliminaries. The second class will be devoted to Python. Please bring your laptop (taught by Jonathan Lindbloom).
Week 3 Chapter 3: Direct Methods for Linear Systems.
Weeks 4 Chapter 4: Least Square Problems.
Week 5 Chapter 7: Polynomial Interpolation.
Week 6 Chapter 8: Approximation of Functions
Week 7 Chapter 9: Differentiation and Integration
Week 8 Chapter 10: Zeroes of Nonlinear Functions
Week 9 Chapter 5: Iterative Methods for Linear Systems

## Course Policies

### Honor Principle

Students are encouraged to work together to understand course material. This includes helping each other by providing insight into homework problems. However, each student is responsible for his/her own assignment, and any homework problem solution that appears to result from a team effort will result in zero points awarded for all parties involved.

### Accessibility Policy

Students needing special accommodations are encouraged to make an office appointment with Professors Gelb and Fu prior to the end of the second week of the term. At this time, students should provide copies of disability registration forms, which list the particular accommodations recommended Student Accessibility Services within the Academic Skills Center. The Director of Student Accessibility is Ward Newmeyer. Office 205 Collis Center; Phone (603) 646-9900.

### Student Religious Observances

Some students may wish to take part in religious observances that fall during this academic term. Should you have a religious observance that conflicts with your participation in the course, please come speak with your instructor before the end of the second week of the term to discuss appropriate accommodations.

### Late Policy

Homework due dates are strictly enforced for full credit. Each day homework is late results in a 10% penalty. Students requesting special accommodations should inform the instructors well in advance so that the instructors will have sufficient time to work with Student Accessibility Services to ensure appropriate accommodation.