MATH 56: 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 or instructor approval. Some experience in MATLAB or another programming language is expected.

Textbooks:

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. 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: 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.
  • First exam: Hand out: TBD. Due: TBD
  • 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). The X hours may be held remotely (check the announcements on CANVAS).
  • Participation & attendance: Students are expected to attend most classes and X hours (when scheduled).
  • Last day of class: March 7 2023
  • Second exam due: TBD

Syllabus

Tentative lecture plan which may be subject to further changes.

Week Lecture
Weeks 1 & 2 Chapters 1 & 2: Preliminaries. The first X-hour will be an introduction to Python in context of the exercises in Chapter 1 (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.