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MATH 6640-100 (10736), Spring 2026
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,
Morton Hall 555.
Office hours (tentatively) Mondays 3:05-4:00pm, Wednesdays 2:00-2:55pm, and Fridays 10:45-11:40am.
Do not hesitate to contact me with questions or to make an appointment to meet at another time.
Monday, Wednesday, Friday 4:10-5:05pm in 314 Morton Hall.
Computational Environment:
We will use the cloud computing
environment CoCalc. Sign
up for a free account, using your @ohio.edu email
address. (You may want to get a paid subscription for better performance.)
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
2025 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.
Attendance:
Your attendance and participation is required and counts toward your grade.
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.
Final Project:
You will individually do a final project to validate (or invalidate) a recent published work. Each student will do a different paper.
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.
Ramp-up Projects:
You will do ramp-up projects on each of the three papers that we cover as a class.
The goal of these projects is to build and practice the skills that will enable you to do your final project well.
You will do these projects individually, but everyone is doing the same thing, so you can help each other (within limits).
Final Exam:
The final exam is scheduled on Monday, April 27, 4:40-6:40pm. There will not be an exam, but your final project notebook is due at that time.
Grade:
Your grade is based on
10% attendance
40% homework
20% ramp-up projects
30% final project
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.
Rules relating to Academic Misconduct:
Any assistance you receive from your classmates, the internet, AI, etc.
must pass through your brain (you may not copy and paste it) and
must be acknowledged, in writing, near where it was used.
You do not need to acknowledge assistance from me.
You may copy and paste
URLs, DOIs, bibliographic information, and similar things meant to be copied;
a direct quote, if you indicate it as a quote and cite its source; and
things from this syllabus and schedule.
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.
You may be absent for up to three (3) days each academic
semester, without penalty, to take time off for reasons of
faith or religious or spiritual belief system or to
participate in organized activities conducted under the
auspices of a religious denomination, church, or other
religious or spiritual organization. You are required to
notify me in writing of specific dates requested for
alternative accommodations no later than fourteen (14) days
after the first day of instruction. These requests will
remain confidential. For more information about this policy,
contact the office of Civil Rights Compliance.
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
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.
This is a very nice book that I previously used as the textbook. However it does not include any developments from the last 25 years and has 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 "Users".
Within "Add new collaborators" search for me mohlenka@ohio.edu and then add me as a collaborator.
Within the project click "Tour" in the upper right and take the tour.
Within the project click "Help" in the upper right and browse the "Features" and "Docs" tabs.
Get our first paper The Generalized Matrix Norm Problem by
Adrian Kulmburg.
SIAM J. Matrix Anal. Appl. 46-4 (2025), pp. 2226-2252, doi:10.1137/23M1605545 (Ohio University proxy link).
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 14
Information:
Our goal today is to learn how to write in Jupyter notebooks using markdown cells and \(\LaTeX\) encoding.
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:
Within your CoCalc project, click "Library" and get
"Markdown in CoCalc" from the
library. Read markdown-intro.md
and markdown-in-jupyter.ipynb.
Create a Jupyter notebook week1.ipynb, selecting the "Python 3 (system-wide)" kernel. 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.
When \(\LaTeX\) is rendered on an html page using MathJax (like this page), you can see the underlying \(\LaTeX\) code by right-clicking on the math, selecting "Show Math As", and then selecting "TeX Commands".
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 definitely need for the paper The Generalized Matrix Norm Problem.
Homework:
Create a Jupyter
notebook week2.ipynb, selecting the "Python 3 (system-wide)" kernel. 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 \(\|AB\|\le \|A\|\, \|B\|\).
To get the documentation on a python function within a jupyter notebook, use "?". So, to get information about range, run range? in a python cell.
Homework:
In a python cell, run
import numpy
from numpy import identity, zeros, dot
from numpy.random import randn
from numpy.linalg import norm
Use randn to generate a random scalar \(a\)
and three random \(10\times 10\) matrices \(A\), \(B\),
and \(C\). Use identity to create the identity
matrix \(I\) and zeros to create the zero matrix \(O\).
Test the following equalities:
\(O+A=A\)
\(A+B=B+A\)
\(A+(B+C)=(A+B)+C\)
\(IA=A\)
\(AB=BA\)
\(A(BC)=(AB)C\)
\(a(B+C)=aB+aC\)
\(A(B+C)= AB+AC\)
(To test an equality like \(A=B\), check \(\|A-B\|\approx 0\).)
Week 3 (January 26)
January 26
Information:
Today was a snow day.
Friday we assigned topics for presentations from the paper The Generalized Matrix Norm Problem:
Section 2.3. Duality.
Definition 2.2 and the statement of Lemma 2.3.
Proof of Lemma 2.3.
Definition 2.4 and the statement of Lemma 2.5.
Proof of Lemma 2.5.
Definition 3.4 and the statement of Lemma 3.6.
Proof of Lemma 3.6, first two paragraphs.
Proof of Lemma 3.6, third paragraph (and lifeline for first two).
Tuesday January 27, 9am: homework week2 due
January 28
Information:
Today we will work on some topics that you identified in the paper The Generalized Matrix Norm Problem.
Homework:
Prepare for your presentation.
Create a Jupyter
notebook week3.ipynb for your work this week.
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 start presentations from the paper The Generalized Matrix Norm Problem.
Week 4 (February 2)
February 2
Information:
Today we will finish presentations from the paper The Generalized Matrix Norm Problem.
Tuesday February 3, 9am: homework week3 due
February 4
Information:
Today we start ramp-up project 1, based on the paper The Generalized Matrix Norm Problem. The goal of project 1 is to practice formatting, background description, goal setting, and similar aspects, without doing the core content of checking the paper's theoretical and numerical claims.
Create a jupyter notebook for your project. In it, fill in (at least) the following:
A title for your project (not the paper title).
Your name, the course, the semester, and the year.
Full bibliographic information for the paper.
An Introduction section with a brief summary of what the paper is about.
A section stating in full Theorem 3.7 and its proof, which is what you would check if this was your final project.
Include definitions for things mentioned, statements of any lemmas used in the proof, etc.
(You may use screenshots from the paper.)
Divide into subsections as appropriate.
A section including Figure 3 and the numerical test it presents, which is what you would check numericaly if this was your final project.
Quote the relevant material from the paper, rather than just referring to the paper.
(You may use screenshots from the paper.)
Divide into subsections as appropriate.
A Conclusions section.
February 6
Information:
Homework:
Get our second paper Matrix Perturbation Analysis of Methods for Extracting Singular Values from Approximate Singular Subspaces by Lorenzo Lazzarino, Hussam Al Daas, and Yuji Nakatsukasa. SIAM J. Matrix Anal. Appl. 46-4 (2025),
pp. 2614-2634, doi:10.1137/24M169343X (Ohio University proxy link).
Make a list of topics it relies on that you do not
know enough about.
Week 5 (February 9)
February 9
Information:
Today we will work on a numerical test related to the paper The Generalized Matrix Norm Problem.
Use Proposition 3.1 to prove the following Corollary:
Let \(\|\cdot\|\) be the matrix 2-norm (largest singular value) and \(\|\cdot\|^*\) be its dual norm (the nuclear norm, the sum of the singular values).
If \(M \in \mathbb{R}^{2\times 3}\) has rank 2,
\(C \in \mathbb{R}^{2\times 4}\),
\(X \in \mathbb{R}^{3\times 4}\) satisfies \(MX=C\),
and \(Y \in \mathbb{R}^{2\times 4}\) satisfies \(\|M^tY\|^*\le 1\),
then \(\|X\| \ge \mathrm{Tr}(C^tY)\).
In a code cell:
Generate random \(M \in \mathbb{R}^{2\times 3}\) and \(C \in \mathbb{R}^{2\times 4}\) and run a test to make sure \(M\) is rank 2.
Use lstsq to generate a matrix \(X_0\) such that \(MX_0=C\).
Use cross to generate a (column) vector \(\mathbf{v}\) in the nullspace of \(M\) with \(\|\mathbf{v}\|=1\) (in Euclidean norm).
Do the rest of your work in other cells, so \(M\) and \(C\) are not re-randomized all the time.
Write a python function randomX that:
Has inputs \(X_0\), \(\mathbf{v}\), and \(m\) with default \(m=4\).
Generates a random vector \(\mathbf{z}\) with \(m\) entries.
Returns \(X=X_0+\mathbf{v}\mathbf{z}^t\).
Run a test to show \(MX=C\).
Write a python function randomY that:
Has inputs \(M\) and \(m\) with default \(m=4\).
Generates a random matrix \(Y_0\) with size \(2\times m\) entries
Returns \(Y=Y_0/\|M^tY_0\|^*\).
Run a test to show \(\|M^tY\|^*=1\).
Run a test that:
Generates many \(X\) satisfying \(MX=C\) and computes the minimal value of \(\|X\|\).
Generates many \(Y\) satisfying \(\|M^tY\|^*=1\) and computes the maximal value of \(\mathrm{Tr}(C^tY)\).
Discuss whether your results validate or contradict the corollary you proved above.
Can you think of a better way to test the corollary?
Tuesday February 10, 9am: homework week4 due
February 11
Information:
Today we will work on some topics that you definitely need for the paper Matrix Perturbation Analysis of Methods for Extracting Singular Values from Approximate Singular Subspaces.
State the formula for the matrix norm induced
by \(\|\cdot\|_2\) in terms of the singular values.
State the formula for the Frobenius norm in terms of
the singular values.
State the formula for the Shatten norm in terms of
the singular values. What is the Shatten quasinorm? What is the nuclear norm?
State the formula for the matrix rank in terms of
the singular values. What can you say about the
singular values of a rank-1 matrix?
Generate a random \(10\times 7\) matrix \(A\), compute its
SVD using numpy.linalg.svd,
and run tests to show the results satisfy the definition
of the SVD.
Thursday February 12, 9am: ramp-up project 1 due
February 13
Information:
Today we will work on some topics that you identified in the paper Matrix Perturbation Analysis of Methods for Extracting Singular Values from Approximate Singular Subspaces.
Homework:
Week 6 (February 16)
February 16
Information:
Today we will work on some topics that you identified in the paper Matrix Perturbation Analysis of Methods for Extracting Singular Values from Approximate Singular Subspaces.
Homework:
Tuesday February 17, 9am: homework week5 due
February 18
Information:
Today we will work on a proof from the paper Matrix Perturbation Analysis of Methods for Extracting Singular Values from Approximate Singular Subspaces.
Homework:
February 20
Information:
Today we will work on a numerical test related to the paper Matrix Perturbation Analysis of Methods for Extracting Singular Values from Approximate Singular Subspaces.
Homework:
Week 7 (February 23)
February 23
Information:
Today we will work on something related to the paper Matrix Perturbation Analysis of Methods for Extracting Singular Values from Approximate Singular Subspaces.
Homework:
Tuesday February 24, 9am: homework week6 due
February 25
Information:
Today we start ramp-up project 2, based on the paper Matrix Perturbation Analysis of Methods for Extracting Singular Values from Approximate Singular Subspaces. The goal of project 2 is to practice the same skills as project 1, and then do small versions of the core content of checking the paper's theoretical and numerical claims.
Create a jupyter notebook for your project. In it, fill in (at least) the following:
A title for your project (not the paper title).
Your name, the course, the semester, and the year.
Full bibliographic information for the paper.
An Introduction section with a brief summary of what the paper is about.
A section stating in full Corollaries 4.2 and 5.1, and then explaining (proving) how they follow from Theorem 4.1. (This is a small version of what you would check if this was your final project.)
Include definitions for things mentioned, statements of any lemmas used in the proof, etc.
(You may use screenshots from the paper.)
Divide into subsections as appropriate.
A section including Figure 2a and the numerical test it presents, and then checking the results with your own implementation.
(This is a small version of what you would check if this was your final project.)
Quote the relevant material from the paper, rather than just referring to the paper.
(You may use screenshots from the paper.)
Divide into subsections as appropriate.
A Conclusions section.
February 27
Information:
Homework:
Get our third paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues by Nils Friess, Alexander D. Gilbert, and Robert Scheichl. SIAM
J. Matrix Anal. Appl. 46-1 (2025),
pp. 626-647, doi:10.1137/23M1622155 (Ohio University proxy link).
Make a list of topics it relies on that you do not
know enough about.
Week 8 (March 2)
March 2
Information:
Today we will work on some topics that you definitely need for the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues.
Write a python function sequencemaker that inputs \(M \gt 0\), \(x_0 \gt 0\), \(q \ge 1 \), and \(N \gt 0\) and outputs the list \([x_0,x_1,\dots,x_N]\) where \(x_k = M x_{k-1}^q\).
What condition is needed on \((M,x_0,q)\) for a sequence generated this way to converge? What will it converge to?
It is claimed that the sequence
\[q_k = \frac{\ln\left(\left|\frac{x_{k+3}-x_{k+2}}{x_{k+2}-x_{k+1}}\right|\right)
}{\ln\left(\left|\frac{x_{k+2}-x_{k+1}}{x_{k+1}-x_k}\right|\right)}\] will converge to the order of convergence of the sequence \(\{x_k\}\).
Write a python function orderestimator that inputs a list \([x_0,x_1,\dots]\) and outputs the list \([q_0,q_1,\dots]\)
Using \((M,x_0,q)=(0.5,10,1)\):
Generate a list using sequencemaker.
Apply orderestimator.
What does it indicate is the order of convergence?
Repeat using \((M,x_0,q)=(0.99,10,1)\), \((1.1,0.5,1.5)\), and \((1.1,0.5,2)\).
Tuesday March 3, 9am: homework week7 due
March 4
Information:
Today we will work on some topics that you definitely need for the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues.
Homework:
Thursday March 5, 9am: ramp-up project 2 due
March 6
Information:
Today we will work on some topics that you identified in the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues.
Homework:
Spring Break
Week 9 (March 16)
March 16
Information:
Your final project is to individually read a paper, do some checks on the proofs, do some numerical tests, and give a presentation. Today we work on finding a suitable paper.
Homework:
Pick two papers from the SIAM Journal on Matrix
Analysis and Applications, published 2023 or later.
They cannot be ones that we have done as a
class. Choose ones that seem interesting and
accessible to you. (Avoid picking the same ones as
your classmates, since the final project papers must
be different.) For each paper:
Upload it to Cocalc.
Make a list of topics it relies on that you do not
know enough about.
Identify 1-3 proofs that you could check.
Identify 1-3 numerical tests that you could do.
Tuesday March 17, 9am: homework week8 due
March 18
Information:
Today we will work on a proof from the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues.
Homework:
March 20
Information:
Today we will work on a numerical test related to the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues.
Homework:
Week 10 (March 23)
March 23
Information:
Today we will work on a numerical test related to the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues.
Homework:
Tuesday March 24, 9am: homework week9 due
March 25
Information:
Today you choose the paper for your project.
Homework:
Select the paper for your project. It can be one of the two you looked at last week or another. Two students cannot do the same paper.
Expand the list of topics it relies on that you do not
know enough about. Include Wikipedia or other web links to them if available.
Give further details on which proof(s) you propose to check. List the main elements of the argument and inlude any you are unfamiliar with in your topic list above.
Give further details on the numerical tests that you propose to do.
Specify which sections of the paper you propose to cover.
March 27
Information:
Today we will work on something related to the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues.
Homework:
Week 11 (March 30)
March 30
Information:
Today we will work on something related to the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues.
Homework:
Tuesday March 31, 9am: homework week10 due
April 1
Information:
Today we start ramp-up project 3, based on the paper A Complex-Projected Rayleigh Quotient Iteration for Targeting Interior Eigenvalues. The goal of project 3 is to practice the same skills as project 2, and then add in a sketch of slides.
This project notebook and slides are due next Thursday.
Create a jupyter notebook for your project. In it, fill in (at least) the following:
A title for your project (not the paper title).
Your name, the course, the semester, and the year.
Full bibliographic information for the paper.
An Introduction section with a brief summary of what the paper is about.
A section stating in full Theorem 3.2 and its proof, and then filling n details and skipped steps in the proof.
Include definitions for things mentioned, statements of any lemmas used in the proof, etc.
(You may use screenshots from the paper.)
Divide into subsections as appropriate.
A section including Figure 2 and the numerical test it presents, and then checking the results with your own implementation.
Quote the relevant material from the paper, rather than just referring to the paper.
(You may use screenshots from the paper.)
Divide into subsections as appropriate.
A Conclusions section.
Upload slides.tex and OHIOCLR.pdf. Edit slides.tex and fill in (at least) the following:
A title for the project.
Your name.
Full bibliographic information for the paper on the slides references page.
An abstract for your talk (not the paper's abstract).
An introduction of 1-3 slides with a brief summary of what the paper is about.
\section{...} and \subsection{...} commands so that the table of contents on page 3 gives an outline of your talk.
April 3
Information:
Homework:
Week 12 (April 5)
April 5
Information:
Homework:
Tuesday April 7, 9am: homework week11 due
April 8
Information:
In week 14 you will do a presentation about your project.
It must be at least TBD and at most TBD+5 minutes long.