# MATH 76.1: Numerical Analysis

Course Information:

Course Description: Numerical analysis is a fundamental topic in applied mathematics. Many practical problems that scientists try to solve are based on mathematical models, but few can be solved analytically, mainly due to their large size. Therefore computational algorithms are needed for approximating these solutions. It is critically important to maintain the important mathematical properties of the underlying system when developing numerical algorithms, and moreover to ensure accuracy, efficiency and convegence. Numerical analysis is about developing good computational techniques for broad based problems and demonstrating that these properties hold both theoretically and computationally. Numerical analysts make sure that computational algorithms are trustworthy so that domain scientists can be confident in the results of their experiments. Numerical analysts also answer the question, ``what assumptions of the underlying problem are necessary for this computational method to succeed?'' 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 are not the main focus of the course.

Prerequisites: Math 22 and Math 23, or per instructor approval. Some experience in MATLAB or another programming language is highly beneficial.

Textbooks:

• Lambers, James V. and Sumner, Amber C. (2018), Explorations in Numerical Analysis, World Scientific Press (required). 2016 preprint of book.
• Ascher, Uri M. and Greif, Chen. (2011) A First Course in Numerical Methods, SIAM (suggested).
The SIAM books are available as e-books through the Dartmouth library. A good reference for MATLAB coding is D.J. Higham and N.J. Higham (2005) MATLAB Guide, Second Edition. SIAM. There are also many other resources online as well as the MATLAB tutorial available through MATLAB.

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

Grading formula: (i) Homework sets (50%); (ii) two take home exams (40%); (iii) Participation & Attendance (10%).

## Important dates and grading information

• First day of class: June 20 2019. NOTE THAT THIS IS THURSDAY X HOUR.
• 5 homework problem sets due approximately every ten days. Homework sets will be handed out in class or 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.
• First exam: Hand out: July 29. Due: August 1 (8:00 am)
• X Hours: Some X hours will be used during the early part of the term, mainly to review class concepts by discussing the ``exploration excercises'' (see textbook). It is a good idea to bring your laptop to X hours.
• Participation & attendance: Students are expected to attend most classes and X hours (when scheduled). I understand that intership interviews often occur during the summer. Please try not to schedule them during class time whenever possible. From time to time during X hours students will have the opportunity to lead discussion on how to approach algorithmic development and/or MATLAB programming. Volunteers are always appreciated, and it's a great way to test your skills.
• Last day of class: August 21 2019
• Second exam due: August 23 2019

## Syllabus

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

Week Lecture
Weeks 1 & 2 Chapters 1 & 2: Preliminaries. The X-hour on June 27 will be for students who want to work through the MATLAB explorations in the book. I will be on hand to answer questions.
Week 3 Chapter 3: Direct Methods for Linear Systems.
Weeks 4 & 5 Chapter 4: Least Square Problems.
Week 6 & 7 Chapter 5: Iterative Methods for Linear Systems.
Week 8 & 9 Chapter 6: Eigenvalue Problems.

## 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.