Math 620 - Numerical Analysis

Fall 2023 - Syllabus - Matthias K. Gobbert

This page can be reached via my homepage at http://www.umbc.edu/~gobbert.

Basic Information

Instructor: Matthias K. Gobbert, gobbert@umbc.edu, office hours Tuesdays and Thursdays 03:00-03:50 in MP 416 in-person, by appointment online, or by e-mail.
Dr. Gobbert has accumulated extensive experience in teaching with state-of-the-art technology. Since participating in the first cohort in the Alternative Delivery Program (ADP) in 2006, he uses hand-writing on tablet laptops for all lectures. These lectures are taped and hosted online for streaming, with the added benefits of allowing for pausing, rewinding, and reviewing. Using the taped lectures for contents delivery, Dr. Gobbert uses a team-based active-learning teaching model, in which students work on problems in learning groups during class. Since 2019, his classes use online comprehension quizzes on the lectures and fully online submission of all assignments, complete with online grading. Since starting online teaching full-time in 2020, the synchronous class meetings are used additionally for student presentations to maximize active student engagement. In 2010, Dr. Gobbert received the University System of Maryland Board of Regents' Faculty Award for Excellence in Mentoring.
This syllabus consists of this frontpage and the sub-pages
- detailed schedule,
- general rules and procedures including grading guidelines, and
- guidelines on how to report on computer results.
- The official ECR website is part of this syllabus.
Classes: Tuesdays and Thursdays, 04:00-05:15, in MP 401 and in Class Collaborate; my intention is to offer the class in HyFlex format, meaning you can attend in-person or online; see the detailed schedule for more information on the coverage, due dates, and timing of the synchronous class meetings.
Note on time commitment: Notice that in a regular semester, you are supposed to spend about 10 to 12 hours per week on this course. If you cannot devote this time, consider if you want to be in this course.
This course will be taught in a flipped classroom format. This means that the contents is delivered asynchronously online by taped videos of my lectures that you study before our synchronous class meetings. The synchronous class meetings will use a team-based active-learning teaching model, in which students work on problems in learning groups formed by the instructor with the help of the instructor.
Learning groups: I will form learning groups of about two to four students. These groups are strongly encouraged to also communicate outside of class.
Synchronous class meetings: Our synchronous class meetings are an opportunity for active teamwork with your learning group, while the instructor is available immediately for questions. These meetings will take place in-person in the classroom as well as optionally in Class Collaborate, which is included with Blackboard, see below and note on recordings. We will also have presentations by students on the recorded lectures and on homework solutions in the synchronous class meetings. This and the other strategies above are designed to foster student engagement as well as give you chances to participate more actively, get to know each other better, have a demonstrated record of using the tools of online learning, and more.
If you have any concerns about any of these items, such as concerns about adequate internet connection, about team work, or special needs related to learning styles, please reach out to me as soon as possible, so I can clarify questions and/or we can work out alternate appropriate approaches and metrics. My goal is definitely that anyone can participate successfully in this course, even if you might need to do it completely asynchronously, but we need to communicate about this.
Prerequisites: a grade of C or better in Math 221 Linear Algebra, Math 225 Ordinary Differential Equations, Math 251 Multivariable Calculus, Math 301 Introduction to Mathematical Analysis, familiarity with a high-level programming language, or consent of instructor.
These books are highly recommended as reference, but are not required. The intention is to cover the material of the course sufficiently well by the lectures, possibly complemented by specific reading assignments, that I will post online.
- Recommended textbook: Kendall E. Atkinson, An Introduction to Numerical Analysis, second edition, Wiley, 1989. Associated webpage: http://www.math.uiowa.edu/~atkinson/keabooks.html including list of errors.
  An analytical introduction to the subject, comprehensive and timeless.
- Recommended book on Matlab/Octave: Desmond J. Higham and Nicholas J. Higham, Matlab Guide, third edition, SIAM, 2017. The associated webpage https://nhigham.com/matlab-guide includes a list of errors, downloadable code, links, and much more. Matlab's documentation is excellent, but along with its functionality has reached a scale that requires a lot of sophistication to fully understand. Moreover, there is a definite role for a book that is organized by chapter on topics such as all types of functions (inline, anonymous, etc.), efficient Matlab programming (vectorization, pre-allocation, etc.), Tips and Tricks, and more.

Grading rule:

Participation	Quizzes	Homeworks	Midterm Exam	Final Exam or Project
10%	15%	25%	25%	25%

Active participation is the key to success in any class, mathematics or otherwise. This category gives you credit for all associated activities including studying the recorded lectures completely, preparing for the synchronous class meetings as assigned, the active participation in team work, the participation in synchronous class meetings, and adequate professional behavior in all aspects of the course, such as communicating with the instructor, team mates, respectful behavior in communications, and timely submission of work, for instance. This grading category will be implemented by Blackboard's automated Attendance tool. To reiterate: As our contents delivery is through my recorded lectures, you are required to study them completely, since they are what a live lecture would be in a lecture format. You are required to study them before class, along with any other reading or preparatory activities, so that we have a common starting point and since you cannot learn actively and with others in the class meetings, if you are not prepared. If you have any concerns about this, see the note above.
The online quizzes are administered in the course management system Blackboard, see below, and are due before class. The detailed schedule indicates the planned due dates of these online quizzes, but the Blackboard assignments list the official due dates and times. There will also be in-class quizzes using learning groups formed by the instructor. For instance, they may be designed to initiate class discussion or to give me feedback on your learning. They may be technical or non-technical in nature. A Quiz 0 will be given on some material that is critical to your success in this class.
The homework assignments will be posted in the Blackboard site of our course, see below. The detailed schedule indicates the planned due date of each homework, but the Blackboard assignments list the official due dates and times. Working the homework is vital to understanding the course material, and you are expected to work all problems. In the flipped classroom format of this course, we will work on the homework about the associated taped lectures on that topic in class. You should then complete the homework afterwards, and it is then due before the start of the next topic. The homeworks are weighted so heavily, because they include the computer assignments that are vital to the computational focus of this course. Homework submission is online as one PDF file in the Assignments area of our Blackboard site. Late submission of homeworks, except Homework 0, cannot be accepted under any circumstances. If homework is accepted late, it accrues a deduction of up to 10% of the possible score for each day late until my receiving it; I reserve the right to exclude any problem from scoring on late homework, for instance, if we discuss it in class. Homework 0 is required of all students and is accepted late.
The midterm and final exams are conventional pencil-and-paper exams. To help you focus on what is relevant, they are closed-book and closed-notes, but you should have a scientific calculator. See the detailed schedule for the dates of the exams.
It is increasingly important at this point in your education to learn how to work on a larger project on your own (with guidance by the instructor) and to present your results in the form of a professional-grade type-set report. To allow interested students to develop the necessary skills, you may choose to replace the final exam by a project; since I assume here that this is a new experience for you and not suitable for all students, you must talk to me for approval. I want to mention that a class project is a great way to start on a research project, in case that is something you are interested in.

Additional details or changes will be announced as necessary. See also general rules and procedures for more information. Announcements may be made in class, by e-mail, or in Blackboard. You are responsible for checking your UMBC e-mail address sufficiently frequently.

We will use the course management system Blackboard Ultra for posting of all material (including homework, lesson plans, PDF transcripts, handouts), for links (to tapings of lectures), and for submission of homeworks. Notice that Blackboard Ultra is the new version of Blackboard that is more mobile friendly. The main navigation buttons are arranged along the top of the screen, with Content and Gradebook the ones we will use. I will also use Blackboard to send Announcements to the class, which goes to your UMBC account by default. Therefore, you must either check your UMBC e-mail regularly or have the mail forwarded to an account that you check frequently. Do not use Blackboard to message me, since I may not find it in a timely fashion; rather use conventional e-mail to my UMBC address listed above.

Course Description

Numerical Analysis is concerned with the approximation of mathematical objects, the analysis of the errors incurred in this approximation, and the development and implementation of computer algorithms for the computation of these approximations. The approximations take various forms including the approximation of a function by a series with finitely many terms or the approximation of a derivative by a finite difference. These approximations incur numerical error, in the examples above known as truncation error and discretization error, respectively.

The methods covered include polynomial interpolation, numerical differentiation and integration, approximation theory and orthogonal polynomials, the solution of systems of non-linear equations, and an introduction to numerical methods for ordinary differential equations. Additionally, we will discuss Gaussian elimination for the solution of systems of linear equations and other selected topics such as the representation of real numbers in computers according to the IEEE-standard for floating-point numbers.

This course will also include computational work to gain practical experience with the numerical methods discussed. I recommend the professional software package Matlab or equivalently the free and nearly fully compatible package Octave as platform of choice, because they are very popular packages and knowing them thoroughly is itself a marketable skill. For both packages, you can read its expansive and well-written documentation or you may consider the book recommended above. For hands-on training in Matlab and Octave, you can consider the 2-credit class Math 426 on Matlab or for a brief initial overview the software workshops offered by CIRC.

Learning Goals

By the end of this course, you should:

understand and remember the key ideas, concepts, definitions, and theorems of the subject. Examples in this course include computational algorithms, sources of error, convergence theorems, and implementations of these algorithms. More broadly, you should also understand the purpose of numerical analysis.
--> This information will be discussed in the lectures as well as in the textbook and other assigned reading.
be able to apply mathematical theorems and computational algorithms correctly to answer questions, and interpret their results correctly, including potentially non-unique solutions or breakdowns of algorithms. Examples include choosing among several methods to solve a problem and how to react to intermediate solutions found that may indicate a breakdown of the method.
--> The class discussions, homework, and quizzes address these skills.
have experience using a professional software package, writing code in it, and understanding how some of its functions work. We will focus on the package Matlab in this course, which is the most popular package in mathematics and many application areas. Writing code in this context includes the requirements to deliver code in a form required, such as writing code to stated specifications, using a requested method, complying with a required function header, etc. The knowledge and skills in this item are valuable job skills, which justifies the emphasis here.
--> This is one of the purposes of the homeworks and most learning will take place there.
have some experience how to learn information from reading and to discuss it. More broadly, communication with peers as well as supervisors is a vital professional skill, and the development of professional skills is a declared learning goal of this course.
--> I will supply some research papers carefully selected for their readability and relevance to the course.
have some foundational experience in writing professional-grade reports in Mathematics. This is explained more in the syllabus portion on How to Report on Computer Results.
--> This is included in the homeworks.
have experience working actively with peers in a group, both on the scale of the class and in a smaller team. Group work requiring communication for effective collaboration with peers and supervisors is a vital professional skill, and the development of professional skills including this networking is a declared learning goal of this course. Additionally, getting to know other students as part of learning groups will prove invaluable for homework and exams.
--> Group discussions and quizzes will contribute to this goal.
have gained experience with a wide variety of teaching and learning techniques, with the goal of encouraging life-long learning.
Examples of teaching techniques include: the use of recorded lectures (that are to be studied before class meetings), the pedagogical technique of a flipped classroom (where we work actively), the user of team-based learning (where you work with peers), and more.
Examples of learning techniques include: learning by reading (e.g., assigned reading before class meetings), learning by hearing (e.g., voice-over in recorded lectures or verbal discussions in class meetings), learning by writing (e.g., answering interpretative questions that require long-hand English answers), learning by talking (e.g., verbal presentations to class), and more.
The reflection on your use of study techniques is a declared learning goal of this course, since it is vital to recognize one's own strengths and weaknesses for best success in live-long learning.
-->All aspects of the course will serve as examples.

Philosophical Underpinning

To provide some context of the more formal learning goals above, I am sharing some deeper thoughts how we fit into the grander scheme of things. The rationale of a state university is to provide a well-educated workforce to the companies in the State of Maryland as well as to the state and local governments themselves. On a fundamental level therefore, you need to able to learn new material as well as have demonstrated evidence of this ability. These are the fundamental purposes of university courses. This requires us to engage in the learning itself and its demonstration; I am trying to say that it is not the solution to a problem that we are after, and in broader thinking it is not even the solution process, but it is your active struggle to learn that we must encourage. This leads me to the following philosophical grade rubric: To earn a passing grade, you need to solve the problem correctly. To earn a good grade, you need to solve the problem correctly and present the solution process completely and professionally. To earn a very good grade, you need to do these and additionally prove that your solution is correct. This rubric comes from the fact that neither you as customer would be willing to accept a faulty solution or unprofessional or incomplete presentation, nor would the companies in the State be served by faulty solutions. These are some of the guiding principles behind my teaching.

Other Information

The recommended literature on my homepage
A brief introduction to Unix/Linux at UMBC
How to get started with Matlab at UMBC (maintained by CIRC)
An introduction to LaTeX gives some pointers how to get started with LaTeX, has a small sample document as well as a larger Template for Project Reports.

UMBC High Performance Computing Facility (HPCF) including general information, list of projects, publications page, and resources for users.
Center for Interdisciplinary Research and Consulting (CIRC)

Note on Recordings and Their Publication

This class is being audio-visually recorded so students who cannot attend a particular session and wish to review material can access the full content. This recording will include students' images, profile images, and spoken words, if their camera is engaged and their microphone is live. Students who do not consent to have their profile or video image recorded should keep their camera off and not use a profile image. Likewise, students who do not consent to have their voice recorded should keep their mute button activated and participate exclusively through alternative formats such as email or the chat feature (where available).

UMBC Statement of Values for Academic Integrity

Academic integrity is an important value at UMBC. By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal. Consult the the UMBC webpage on Academic Integrity at academicconduct.umbc.edu for the UMBC Undergraduate Student Academic Conduct Policy for undergraduate students and the UMBC Graduate School's Policy and Procedures for Student Academic Misconduct for graduate students.