University of Chicago

Fall 2017

This is an introductory course on numerical linear algebra. The course will present a global overview of a number of topics, from classical to modern to state-of-the-art. The fundamental principles and techniques will be covered in depth but towards the end of the course we will also discuss some exciting recent developments.

Numerical linear algebra is quite different from linear algebra. We will be much less interested in algebraic results that follow from the axiomatic definitions of fields and vector spaces but much more interested in analytic results that hold only over the real and complex fields. The main objects of interest are real- or complex-valued matrices, which may come from differential operators, integral transforms, bilinear and quadratic forms, boundary and coboundary maps, Markov chains, graphs, metrics, correlations, hyperlink structures, cell phone signals, DNA microarray measurements, movie ratings by viewers, friendship relations in social networks, etc. Numerical linear algebra provides the mathematical and algorithmic tools for matrix problems that arise in engineering, scientific, and statistical applications.

- 11/28/17: Office hours today cancelled. Send questions by email.

- 11/26/17: Class on Nov 27 cancelled.

- 11/21/17: Lecture notes 18 posted.

- 11/20/17: Homework 5 and Lecture notes 17 posted.

- 11/13/17: Lecture notes 16 posted.

- 11/08/17: Homework 4 and Lecture notes 15 posted.

- 11/06/17: Lecture notes 14 posted.

- 11/01/17: Lecture notes 13 posted.

- 10/30/17: Homework 3 and Lecture notes 12 posted.

- 10/24/17: Reminder: Quiz I on Wed, Oct 25, in class.

- 10/23/17: Lecture notes 11 posted.

- 10/19/17: Homework 2 and Lecture notes 10 posted.

- 10/16/17: Lecture notes 9 posted.

- 10/13/17: Lecture notes 8 posted.

- 10/12/17: Greg will hold office hours next Wed, Oct 18, 8:30–10:30am, Jones 304.

- 10/11/17: Reminder: Make-up lecture this Fri, Oct 13, 3:00–4:20pm, Stuart 102.

- 10/11/17: Lecture notes 7 posted.

- 10/09/17: Video of first and second make-up lectures available here (see Sep 30 email for password).

- 10/09/17: Homework 1 and Lecture notes 6 posted.

- 10/06/17: Lecture notes 5 posted.

- 10/05/17: Greg will hold office hours this Fri, Oct 6, 12:00–2:00pm, Jones 304.

- 10/04/17: Reminder: Make-up lecture this Fri, Oct 6, 3:00–4:20pm, Stuart 105.

- 10/04/17: Lecture notes 4 posted.

- 10/02/17: Lecture notes 3 posted.

- 09/30/17: Lecture notes 2 posted. Video of first make-up lecture available here.

- 09/27/17: Lecture notes 1 and Homework 0 posted.

- 09/27/17: First two make-up lectures on Fri, Sep 29 and Oct 6, 3:00–4:20pm, in Stuart 105. Third make-up lecture on Fri, Oct 13, 3:00–4:20pm, in Stuart 102.

- 09/26/17: Class will meet for the first lecture on 3:00pm, Wed, Sep 27, in Stuart 105.

- 09/26/17: Check back regularly for announcements.

**Location:** Room 105, Stuart
Hall.

**Times:** 3:00–4:20pm on Mon and Wed.

**Instructor:** Lek-Heng
Lim

Office: Jones 122B

`lekheng(at)galton.uchicago.edu`

Tel: (773) 702-4263

**Office hours:** 3:00–5:00pm on Tue.

**Course Assistant:** Greg
Naitzat

Office: Jones 203/204

`gregn(at)galton.uchicago.edu`

**Office hours:** Two hours on the weekday before a problem set is due
(time/venue to be announced).

The last two topics we would only touch upon briefly (no discussion of actual algorithms); they would be treated in greater detail in a second course.

- Linear algebra over
**R**or**C**: How this course differs from your undergraduate linear algebra course.

- Three basic matrix decompositions: LU, QR, SVD.

- Gaussian elimination revisited: LU and LDU decompositions.

- Backward error analysis: Guaranteeing correctness in approximate computations.

- Gram–Schmidt orthogonalization revisited: QR and complete orthogonal decompositions.

- Solving system of linear equations in the exact and the approximate sense: Linear systems, least squares, data least squares, total least squares.

- Low rank matrix approximations and matrix completion.

- Iterative methods: Stationary methods and Krylov subspace methods.

- Eigenvalue and singular value problems.

- Sparse linear algebra: Sparse matrices and sparse solutions.

Collaborations are permitted but you will need to write up your own solutions and declare your collaborators. The problem sets are designed to get progressively more difficult. You will get about 10 days for each problem set.

- Problem Set 5 (posted: Nov 20, due: Nov 29)

- Problem Set 4 (posted: Nov 8, due: Nov 20)

- Problem Set 3 (posted: Oct 30, due: Nov 8)

- Problem Set 2 (posted: Oct 19, due: Oct 30)

- Problem Set 1 (posted: Oct 9, due: Oct 18)

- Problem Set 0 (posted: Sep 27, due: Oct 9)

**Bug report** on the problem sets:
`lekheng(at)galton.uchicago.edu`

- Course homepage from Fall 2009 (courtesy of Yali Amit), Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2016. Related course homepages from Fall 2005 and Spring 2006.

**Grade composition:** 50% Problem Sets (six altogether, lowest
grade would be dropped), 50% Quizzes (two altogether, in-class, closed
book)

**Exam dates:** Quiz I on Wed, Oct 25. Quiz II on Wed, Nov 29.

We will use the 4th edition of Golub–Van Loan.

- D.S. Bernstein, Matrix Mathematics, 2nd Ed., Princeton, 2009.

- J. Demmel, Applied Numerical Linear Algebra, SIAM, 1997.

- G. Golub, G. Meurant, Matrices, Moments and Quadrature with Applications, Princeton, 2010.

- G. Golub, C. Van Loan, Matrix Computations, 4th Ed., John Hopkins, 2013.

- N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd Ed., SIAM, 2002.

- M. Overton, Numerical Computing with IEEE Floating Point Arithmetic, SIAM, 2001.

- R. Thisted, Elements of Statistical Computing: Numerical Computation, CRC, 1988.

- L.N. Trefethen, D. Bau, Numerical Linear Algebra, SIAM, 1997.