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MATH 6640-100 (11486), Spring 2019

Numerical Analysis: Linear Algebra

Syllabus

Catalog Description:

In-depth analysis of numerical aspects of problems and algorithms in linear algebra.

Desired Learning Outcomes:

Students will have a deep understanding of numerical methods
for linear algebra. They will know the standard methods and be able
to analyze and learn new methods on their own.

Prerequisites:

MATH 5600 Introduction to Numerical Analysis

Instructor:

Martin J. Mohlenkamp,
mohlenka@ohio.edu,
(740)593-1259, 315-B Morton Hall.
Please communicate with me using your @ohio.edu email address. I will not send any private information to other addresses.

Office hours:

Monday, Wednesday, and Friday 2:00-2:55pm.

By appointment. Do not hesitate to ask for an appointment.

Monday, Wednesday, Friday 3:05-4:00pm in 314 Morton Hall.

How this courses is structured:

In most courses, the material is developed from basic to
more advanced and keeps building. That order is nice and
logical, but means that the material is often poorly motivated and
the course never gets to current research in the area.

This course is organized backwards. We will start with
papers on numerical linear algebra that were published in
2018 and then learn whatever background material we need in
order to understand them. This order is illogical and
chaotic, but much closer to how mathematics is learned and
used in research and professionally.

Homework:

We will have some homework due each week, but the size may vary.
Types of homework will include:

computer calculations,

problem solving and proofs,

reading tasks, such as reading a paper and formulating questions about it,

writing tasks, such as answering questions about a paper you have read or describing a background concept, and

short presentations.

On the writing tasks we will also work on writing skills using the
Good Problems program.

Tests:

We will have three tests. They will test definitions and proofs,
rather than calculations.

Final Project:

You will individually do a final project to validate (or invalidate) a recent published work.
You will produce:

An oral presentation.

Presentation slides, in \(\LaTeX\) using the beamer
class.

A Jupyter notebook with:

Analytic checks of the paper, such as filling in missing steps in proofs, checking that results cited from other papers are really in those papers, etc.

Numerical checks of the paper, such as reproducing their numerical results, using simulations to test inequalities, etc.

Final Exam:

The final exam is scheduled on Wednesday, May 1, 12:20-2:20pm. There will not be an exam, but your final project notebook is due at that time.

Attendance:

Your
attendance and participation is essential for the operation of this class. You
are allowed 4 absences (out of 41 classes) without penalty;
these include university excused absences for illness, death in
the immediate family, religious observance, jury duty, or
involvement in University-sponsored activities. Each additional
absence will reduce your final average by 0.5%. Your attendance
record will be available in Blackboard.

Grade:

Your grade is based on homework at 40%, tests at
30%, and the final project at 30%. An average
of 90% guarantees you at least an A-, 80% a B-, 70% a C-, and
60% a D-. Grades are not the point.

Computational Environment:

We will use the cloud computing
environment CoCalc. Sign
up for a free account, using your @ohio.edu email
address.

Academic Misconduct:

Homework and Project:

These must be your own work, but you can use any help
that you can find as long as you acknowledge in writing what
help you got and from whom or where. (You do not have to
acknowledge me.) Your words must be your own; any text not
your own must be properly quoted and cited.

Tests:

You may not give or receive any
assistance during a test, including but not
limited to using notes, phones, calculators, computers, or
another student's solutions. (You may ask me questions.) A
minor, first-time violation will result in a zero grade on
that test.

Serious or second violations will result in failure in the
class and be reported to the Office of
Community Standards and Student Responsibility, which may
impose additional sanctions. You may appeal any sanctions
through the grade appeal process.

Special Needs:

If you have specific physical,
psychiatric, or learning disabilities and require
accommodations, please let me know as soon as possible so that
your learning needs may be appropriately met. You should also
register with Student Accessibility
Services to obtain written documentation and to learn about
the resources they have available.

Responsible Employee Reporting Obligation:

If I learn of any instances of sexual harassment, sexual
violence, and/or other forms of prohibited discrimination, I
am required to report them. If you wish to share such
information in confidence, then use one of the confidential
resources listed by
the Office
of Equity and Civil Rights Compliance.

Learning Resources:

People:

Your classmates are your best resource. Use them!

Books:

Numerical Linear Algebra,
by Lloyd N. Trefethen and David Bau III.
Society for Industrial and Applied Mathematics, 1997;
ISBN 978-0-898713-61-9. (Become a student member of SIAM and
buy it
through them at a discount.)
This is a very nice book that I used as the textbook when I last taught this course. I am not using it now because it does not include any developments in the last 20 years and because of posted homework solutions.

Internet searches will reveal many other sources and
copies of books. If we find some particularly useful
(and not copyright violations), then I will add them to
this list.

Sign up for a free account, using
your @ohio.edu email address.

Create a project. Within it hit "Settings". Within "Add new collaborators" search for me mohlenka@ohio.edu and then add me as a collaborator.

Within the project click "New", from the library select "First Steps in CoCalc" and hit "Get a Copy". Click the file first-steps.tasks. Do the tasks up to and including the Jupyter Notebook.

Get our first paper Concave Mirsky Inequality and Low-Rank Recovery by
Simon Foucart.
SIAM J. Matrix Anal. Appl. 39-1 (2018), pp. 99-103, doi:10.1137/16M1090004. Upload it into CoCalc. Read it by the next class and make a list of topics it relies on that you do not know enough about.

January 16

Our goal today is to learn how to write in Jupyter notebooks using markdown cells and \(\LaTeX\) encoding.

Within your CoCalc project, click "New" and get
"Markdown in CoCalc" from the
library. Read markdown-intro.md
and markdown-in-jupyter.ipynb.

Create a Jupyter notebook week1.ipynb. Put
your name in a markdown cell.

Put the list of topics that you made about the paper in
another markdown cell.

For each of the following topics, make a markdown cell
with a brief (e.g. 1-3 sentences and a formula) summary and
links/citations to the sources you used. Use \(\LaTeX\)
encoding for any formulas.

Today we continue learning how to write and refresh our
knowledge of definitions.

Read more
about markdown
syntax. In particular, learn about blockquotes;
use them whenever you have copied something into
your homework.

For each of the following topics, make a markdown cell
with a brief (e.g. 1-3 sentences and a formula) summary and
links/citations to the sources you used. Use \(\LaTeX\)
encoding for any formulas.

Today we will work on some topics that you identified in the paper.

In \(\LaTeX\) either $...$
or \(...\) can be used to indicate inline
math. The form \(...\) is better within
html using MathJax, but markdown works better
with $...$. Similarly, markdown
prefers $$...$$ over \[...\]
for displayed math.

Homework:

Create a Jupyter
notebook week2.ipynb. Put your name in a
markdown cell. Do your work this week in it.

Read
about Norm
and Matrix
norm. Using these or other sources, answer the
following, each in a markdown cell. Remember to cite
your sources.

State the defining properties of a norm.

Let \(f(\mathbf{x})\) be the function defined
on vectors \(\mathbf{x}=(x_1,x_2,x_3)\in
\mathbb{R}^3\) by
\[f(\mathbf{x})=|x_1|+2|x_2|+3|x_3|\,.\] Determine
whether or not \(f\) is a norm.

Define what it means for a matrix norm to be
induced by a vector norm.

Show that if \(A\) and \(B\) are square matrices
and the matrix norm is induced by a vector norm,
then \(\|AB\|\le \|A\|\, \|B\|\).