January 14

• Introduction, syllabus, etc.
• Get set up on CoCalc:
  • Use Chrome (or Firefox), not Internet Explorer.
  • 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.
• Look at the \(\LaTeX\) Wikibook for math encoding.
• 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.
  • Vector
  • Vector space
  • Span
  • Inner (dot) product
  • Orthogonal vectors
  • Outer product

January 18

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.
  • Matrix
  • Matrix range
  • Matrix nullspace
  • Rank
  • Invertible matrices
  • Orthogonal matrix
  • Unitary matrix

January 21

Martin Luther King Jr. day holiday, no class

Tuesday January 22, 9am: homework week1 due

January 23

Information:

• 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 \(\|AB\|\le \|A\|\, \|B\|\).
• State the defining properties of a seminorm.
• State the defining properties of a quasinorm.

January 25

Information:

• Today we will work on some topics that you identified in the paper.
• We will also go through an existence proof for the SVD.
• 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:

• Read about the Singular Value Decomposition (SVD).
  • State its definition.
  • 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?

January 28

Information:

• Today we get used to programming in python within jupyter notebooks within CoCalc.
• You will need to know about NumPy and its functions randn, identity, zeros, dot, and norm.

Homework:

• Create a Jupyter notebook week3.ipynb for your work this week. Make sure it is using a python kernel.
• 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 \(\|A-B\|\approx 0\).)

Tuesday January 29, 9am: homework week2 due

January 30

Information:

• Today we work more on coding and test the classical Mirsky inequality.
• 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:

• 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.
• Write a program that inputs the size \(n\), generates random matrices \(X\) and \(Y\), computes both sides of the classical Mirsky inequality as stated in the paper, and reports whether or not the inequality was satisfied. Test it.
• Get our second paper A Generalized Matrix Inverse That Is Consistent with Respect to Diagonal Transformations by Jeffrey Uhlmann. SIAM J. Matrix Anal. Appl. 39-2 (2018), pp. 781-800, doi:10.1137/17M113890X. We will only read sections 1. and 2., ending on page 786. Make a list of topics it relies on that you do not know enough about.

February 1

Today we will start going through the proof of Theorem 1 in the first paper.

February 4

Information:

• Today we will continue going through the proof of Theorem 1 in the first paper.

Homework:

• After we finish the proof together, you will be assigned one of the following parts of the proof:
  • Step 1a. through "... \((t_m,f(t_m))\)."
  • Step 1a. from "By concavity of \(f\),..."
  • Step 1b.
  • Step 2.
  • Step 3.
• As homework, prepare to present that part of the proof.

Tuesday February 5, 9am: homework week3 due

February 6

Today you will each have 10 minutes to present as much of your part of the proof of Theorem 1 as you can. At the end I will announce which part will be on the test.

February 8

Test. The test will be on paper, without computers, and consist of:

• Stating some of the definitions and formulas you have done in your homework.
• Proving the existence theorem for the SVD when the matrix is square and invertible. You will be given the spectral theorem.
• Proving some part of Theorem 1.

February 11

Information:

• Today we practice with python and check the concave Mirsky inequality.

Homework:

• Read about lists, the for statement, range, the if statement, and defining functions
• Write python functions for \(g(x)=x\), \(s(x)=\sqrt{x}\), \(h(x)=\min\{1,x\}\), and \[p(x)=\begin{cases}4x & \text{if \(x \le 1\)}\\ 4+3(x-1) & \text{if \(1 \lt x \le 2\)}\\ 7+2(x-2) & \text{if \(2 \lt x\)}\end{cases}\,.\]
• Write a program that inputs sizes \(n_1\) and \(n_2\) and function \(f\), generates random \(n_1\times n_2\) matrices \(X\) and \(Y\), computes both sides of the concave Mirsky inequality as stated in Theorem 1 in the paper, determines whether or not the inequality was satisfied, and returns True or False.
• Using nested for loops, test all combinations with f in [g,s,h,p], n1 in [2,10,100], and n2 in [2,5,15,20].

February 13

Information:

• Today we get started on definitions from the second paper.
• Each of you will be assigned a proof bit, which you will present next week:
  • Proof of Lemma 2.2
  • Proof of Theorem 2.3, Equations (39) to (44)
  • Proof of Theorem 2.3, Equations (45) to (48)
  • Proof of Theorem 2.3, Equations (49) to (50)
  • Proof of Theorem 2.3, Equations (51) to (60)

Homework:

• Write the definition for each of the following. Include citations of your sources.
  • Nilpotent matrix
  • Idempotent matrix
  • Permutation matrix
  • Conformable matrix
  • Positive Definite Matrix
  • Drazin inverse
  • Polar decomposition
• Read about numpy.random.permutation, scipy.linalg.polar
• Make a python function that returns a random \(n\times n\) permutation matrix \(P\). Run a test to show \(P\) is orthogonal.
• Generate a random matrix, compute its polar decomposition, and show that the results satisfy the definition of the polar decomposition.

February 15

Information:

• Today we learn about the Moore-Penrose pseudoinverse and reproduce an example from the paper.

Homework:

• Read about the Moore-Penrose pseudoinverse. State its definition.
• Use numpy.linalg.pinv to compute the pseudoinverse and run tests to show it satisfies the definition.
• Use the SVD of \(A\) produced by svd() to construct the pseudoinverse of \(A\) and compare with the one produced by pinv().
• Reproduce the example in the paper in Equations (13) to (16).

February 18

Information:

• Today we first discuss Unit Consistency.
• Then you will do some coding in preparation for Definition 2.1 and Lemma 2.2
• The person assigned to prove Lemma 2.2 should be ready for Wednesday and the others should be ready for Friday.

Homework:

• Make a python function that returns a random \(n\times n\) diagonal matrix \(D_+\) with positive entries on the diagonal.
• Make a python function that returns a random \(n\times n\) diagonal matrix \(D_u\) that is also real and unitary.
• Make a python function that returns a random \(n\times n\) unitary matrix \(U\). (Hint: use the output of svd().)
• Make a python function that has a matrix \(A\) as input and returns the matrix \(D\) in equation (29). Use \(\|\cdot\|_2\) as the vector norm.

Tuesday February 19, 9am: homework week5 due

February 20

Information:

• Today we go through the statement and proof of Lemma 2.2, with one of you presenting the proof.
• Then we do numerical tests to verify Lemma 2.2.

Homework:

• Write tests to check Lemma 2.2. Specifically, show that the matrix \(D\) in equation (29) satisfies equations (25)-(28). Test for both square and rectangular \(A\) and with some \(A\) that have a zero row.
• Get our third paper Subspace Acceleration for the Crawford Number and Related Eigenvalue Optimization Problems by Daniel Kressner, Ding Lu, and Bart Vandereycken. SIAM J. Matrix Anal. Appl. 39-2 (2018), pp. pp. 961-982, doi:10.1137/17M1127545. We will only read through section 2.1, ending on page 966. Make a list of topics it relies on that you do not know enough about.

February 22

Information:

• Today we go through the statement and proof of Theorem 2.3, with four of you presenting the proof.
• Then we do numerical tests to verify Theorem 2.3.

Homework:

• Write tests to check Theorem 2.3.

February 25

Today we catch up

Tuesday February 26, 9am: homework week6 due

February 27

Information:

• Today we learn connections between the SVD, pseudoinverse, and least-squares problems.
• For a linear system \(Ax=b\), the residual is defined by \(r=b-Ax\). A least-squares solution is an \(x\) such that \(\|r\|_2\) is minimized. One goal today is to show that \(x = A^+b\) is a least-squares solution, where \(A^+\) is the (Moore-Penrose) pseudoinverse.

Homework:

• Prove that \(x\) is a least-squares solution to \(Ax=b\) if and only if it is a least-squares solution to \((UA)x=(Ub)\) for any conformable unitary matrix \(U\).
• Prove that \(x\) is a least-squares solution to \(Ax=b\) if and only if \(y=U^*x\) is a least-squares solution to \((AU)y=b\) for any conformable unitary matrix \(U\).
• Prove that if \(A\) is a diagonal (not necessarily square) matrix then \(x = A^+b\) is a least-squares solution to \(Ax=b\).
• Using the above and the definiton of the pseudoinverse in terms of the SVD, show that \(x = A^+b\) is a least-squares solution to \(Ax=b\) for any \(A\).
• Run numerical tests of the claim: \(x\) is a least-squares solution to \(Ax=b\) if and only if \(A^*r=0\). Test for both square and rectangular \(A\) and with some \(A\) that have a zero row or a zero column.

March 1

Information:

• Today we start on definitions from the third paper and related coding.
• Each of you is assigned a proof bit from our third paper, which you will present after spring break:
  • Proof of Lemma 2.1 (a)
  • Proof of Lemma 2.1 (b) through "obtain \(\phi(\theta;V)=\phi(\theta)\)."
  • Proof of Lemma 2.1 (b) from "If \(\phi(\theta)\) is a simple".
  • Proof of Theorem 2.4 through \(s_1=s_0+\ln \varepsilon_1 \lt 0\).
  • Proof of Theorem 2.4 starting "By standard techniques..."

Homework:

• State the definitions of the Rayleigh Quotient and the numerical range.
• State the Hausdorff-Toeplitz Theorem and explain the terms in it.
• Write a python function that inputs \(n\) and returns a random matrix in \(\mathbb{C}^{n\times n}\). (Hint: Make two real matrices and add them as A1+1j*A2.)
• Write a python function that inputs a square matrix \(A\) and an integer \(N\) and returns \(N\) random points in the numerical range of \(A\).
• Generate a random matrix in \(\mathbb{C}^{4\times 4}\), compute 10000 random points in its numerical range, and plot the points using matplotlib.pyplot.plot. Compare with Figure 1 in the paper.

March 4

Information:

• Today we experiment with the Crawford number.

Homework:

• Write a python function AtoSK that inputs \(A \in \mathbb{C}^{n\times n}\) and returns the matrices \(S\) and \(K\) as used in Equation (2).
• Write a python function mineigSKtheta that inputs hermitian matrices \(S\) and \(K\) in \(\mathbb{C}^{n\times n}\) and \(\theta \in [0,2\pi]\) and returns \(\lambda_{\mathrm{min}}(S \cos(\theta)+K\sin(\theta))\) as in Equation (2). (Use numpy.linalg.eigvalsh to get the eigenvalues.)
• Generate a random matrix in \(\mathbb{C}^{4\times 4}\), compute \(S\) and \(K\), run mineigSKtheta for 100 values of \(\theta \in [0,2\pi]\), and plot the resulting \(\lambda_{\mathrm{min}}\) as a function of \(\theta\). Compare with Figure 1 in the paper. Based on your plot, what is the Crawford number of your \(A\)?

Tuesday March 5, 9am: homework week7 due

March 6

Information:

• Today we work on some topics that you identified in the paper.
• First, I will discuss optimization terms: Objective function, Univariate optimization, Eigenvalue optimization
• Then we will discuss the main ideas of low-dimensional surrogate models, subspace methods, subspace acceleration, and monotonicity.

Homework:

• State the Min-max Theorem, which gives a variational characterization of the eigenvalues of a Hermitian matrix.
• State the Cauchy interlacing theorem. (Include all assumptions in the theorem statement.)

March 8

Test. The test will be on paper, without computers, and consist of:

• Stating some of the definitions and formulas you have done in your homework in weeks 5-7.
• Formulas and proofs related to the Moore-Penrose pseudoinverse, such as:
  • State the defining properties of the Moore-Penrose pseudoinverse \(A^+\).
  • State the formula for \(A^+\) using the SVD of \(A\).
  • Show that the \(A^+\) defined using the SVD satisfies the properties of the pseudoinverse.
  • Prove that \(x=A^+b\) is a least-squares solution of \(Ax=b\).
• Formulas and proofs from the second paper, such as:
  • Explain the similarities and differences between a unitary consistent generalized matrix inverse and a unit consistent generalized matrix inverse.
  • From Lemma 2.2, prove that (29) satisfies (25), (26), and (28).
  • From Theorem 2.3, prove that (34) satisfies (35) and (36).

March 18

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, volume 39 (2018), which has issues 1, 2, 3, and 4. 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 19, 9am: homework week8 due

March 20

Information:

• Today we first go through the proof of Lemma 2.1(a), with one of you presenting.
• Then we do a simple check on Lemma 2.1(a).

Homework:

• Make a python function that inputs inputs hermitian matrices \(S\) and \(K\) in \(\mathbb{C}^{n\times n}\), a matrix \(V \in \mathbb{C}^{n\times k} \) with orthonormal columns, and a scalar \(\theta \in [0,2\pi]\), and returns \(\lambda_{\mathrm{min}}(V^*(S \cos(\theta)+K\sin(\theta))V)\). This is called \(\phi(\theta;V)\) in the paper.
• Generate a random matrix \(A\in\mathbb{C}^{4\times 4}\) and compute its \(S\) and \(K\) (as you did in week 8). Generate a random unitary matrix \(W\in\mathbb{C}^{4\times 4}\). Let \(U\in\mathbb{C}^{4\times 2} \) be the first two columns of \(W\) and \(V\in\mathbb{C}^{4\times 3} \) be the first three columns of \(W\).
• Plot \(\phi(\theta;U)\), \(\phi(\theta;V)\), and \(\phi(\theta)\) on the same axis to visually test Lemma 2.1(a).

March 22

Information:

• Today we first go through the proof of Lemma 2.1(b), with 2 of you presenting.
• Then we will do a simple numerical test of Lemma 2.1(b).

Homework:

• Generate a random matrix \(A\in\mathbb{C}^{4\times 4}\) and compute its \(S\) and \(K\) (as you did in week 8). Let \(B=S \cos(1)+K\sin(1)\). Use numpy.linalg.eigh to compute the eigenvalues and eigenvectors of \(B\) and let \(V \in\mathbb{C}^{4\times 2}\) be the matrix whose columns are the eigenvectors of \(B\) with smallest eigenvalues.
• Plot \(\phi(\theta;V)\) and \(\phi(\theta)\) on the same axis. What does Lemma 2.1(b) say about these two graphs? Do your graphs agree with the Lemma?

March 25

Information:

• Today we work on the concept fo rate of convergence.

Homework:

• Read about Rate of convergence. State the definitions for:
  • linear convergence
  • sub-linear convergence
  • super-linear convergence
  • quadratic convergence
  • order of convergence
  • Q-convergence
  • R-convergence
• 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 26, 9am: homework week9 due

March 27

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 29

Information:

• Today we first go through Algorithm 1,
• Then we will go through the statement of Theorem 2.3
• Then we will go through the proof of Theorem 2.4, with 2 of you presenting.

April 1

Information:

• First we finish up any proofs from Friday.
• Then we will do a simple numerical test of Theorem 2.4.

Homework:

• Make a python function that inputs \(e_0\), \(e_1\), \(C\), and \(K\) and returns \([e_0,e_1,e_2,\dots,e_K]\) with \(e_{k+1}\) determined by equation (7) with \(=\) instead of \(\le\).
• Use your function to generate a sequence with \((e_0,e_1,C,K)=(0.4,0.3,2,12)\).
• Apply orderestimator. What does it indicate is the order of convergence? Does it agree with Theorem 2.4?

Tuesday April 2, 9am: homework week10 due

April 3

Information:

• Today we get started on the files for your project.

Homework:

• Create a jupyter notebook for your project. In it, fill in (at least) the following:
  • A title for the project.
  • Your name, the course, the semester, and the year.
  • Full bibliographic information for the paper.
  • An introductory section with a brief summary of what the paper is about.
  • A section stating in full the theorems, lemmas, etc. whose proofs you will check. Include definitions for things mentioned. Divide into subsections as appropriate.
  • A section stating the things you will test numericaly. Quote the relevant material from the paper, rather than just refering to the paper. Divide into subsections as appropriate.
• 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 5

Information:

• Today we learn about condition number.

Homework:

• Read about Condition Number. Summarize in your own words. State the relationship between condition number and the error in solving a linear system. State the formula for the condition number of a matrix with respect to \(\|\cdot\|_2\) in terms of the singular values.
• Write a python function with inputs \((m,n,k)\) that returns a random-looking matrix \(A\) such that
  • \(A \in \mathbb{C}^{m\times m}\),
  • \(\|A\|_2=n\), where \(\|A\|_2\) is the matrix norm induced by the vector norm \(\|\cdot\|_2\), and
  • the condition number of \(A\) with respect to \(\|\cdot\|_2\) is \(k\).
• Validate your function using A.shape, numpy.linalg.norm(A,ord=2), and numpy.linalg.cond(A,p=2).
• Write a python function with inputs \((m,n,k)\) that calls your function above to create \(A\), creates a \(m\times 1\) vector \(y\), computes \(b=Ay\), solves \(Ax=b\) for \(x\) using numpy.linalg.solve, and returns the relative error \(\|x-y\|_2/\|y\|_2\).
• State the theoretical dependence of that relative error on \(m\), \(n\), and \(k\). Use your function to test this dependence. Include the case \((m,n,k)=(50,10^3,10^{14})\) in your tests.

April 8

Information:

• Today we work on the concept of computational cost.
• Read about Analysis of algorithms.
• Read about the python timeit library.

Homework:

• In theory, the computational complexity of adding two vectors in \(\mathbb{R}^n\) is \(\mathcal{O}(n)\). The following code tests this.

  import timeit
  # setup, not timed
  def s(n): return "import numpy as np; x=np.random.randn(%s); y=np.random.randn(%s)"%(n,n)
  # operation to test
  op = "z=x+y"
  # sizes to try
  ns = [2**j for j in range(20)]
  # theoretical order of cost
  def c(n): return n
  # number of repetitions to use
  def r(n): return 2**21/c(n)
  # time to do the operation
  T = [timeit.timeit(op,setup=s(n),number=r(n))/r(n) for n in ns]
  # time compared to theoretical order
  Tc = map(lambda t,n: t/c(n), T, ns)
  for i in range(len(ns)):
      print i,ns[i],T[i],Tc[i]

  Run this code and interpret the results.
• We would also like to test the following operations:
  • Adding two square matrices.
  • Multiplying a square matrix times a vector.
  • Multiplying two square matrices.
  • Computing the SVD of a square matrix.
  For each of these, modify the above code, run a test, and interpret the results.
• How does the time it takes to compute the SVD compare to the time for other operation(s) that have the same theoretical order?

Tuesday April 9, 9am: homework week11 due

April 10

Information:

• Today is for working on your project.
• In week 14 you will do a presentation about it.
  • It must be at least 20 and at most 25 minutes long.
  • It must use the \(\LaTeX\) slides template you started last week.
  • Aim for half the presentation to explain to the class what the paper is about and half to show what you did.
  • It is worth 20% of your project grade.
  • You will rate each other
• If you would like to do a practice talk next week, then let me know.

April 12

Test: The test will be on paper, without computers, and consist of:

• Stating some of the definitions and formulas you have done in your homework in weeks 8-11.
• Material from the third paper.
  • Lemma 2.1.
    • State the lemma.
    • Explain in words and graphically what it means.
    • (You do not need to prove it.)
  • Algorithm 1.
    • Write down the algorithm.
    • Explain what each line does. Use graphs to illustrate.

April 15

Information:

• First we set the order for the presentations next week.
• Then we learn a little about some common matrix decompositions.

Homework:

• Read about the QR Decompositon, state its definition, and state one use for it. Generate some random \(A \in \mathbb{C}^{m\times m}\), get their QR decomposition using numpy.linalg.qr, and show that the results satisfy the definition.
• Read about the LU Decompositon, state its definition, and state one use for it. Generate some random \(A \in \mathbb{C}^{m\times m}\), get their LU decomposition using scipy.linalg.lu, and show that the results satisfy the definition.
• Read about the Cholesky Decompositon, state its definition, and state one use for it. Generate some random positive definite \(A \in \mathbb{C}^{m\times m}\), get their Cholesky decomposition using numpy.linalg.cholesky, and show that the results satisfy the definition.

April 17

Today is time for working on projects.

Thursday April 18, 9am: homework week12-13 (through April 15) due

April 19

Information:

• First I will discuss Stability in numerical linear algebra.
• Then there will be some time for working on projects.

April 22

Two presentations for the final project.

April 24

Two presentations for the final project.

April 26

One presentation for the final project. Clean up.

Wednesday, May 1

Final Exam 12:20-2:20pm. Your final project notebook is due at 2:20pm.