University of Chicago

Fall 2018

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.

- 12/04/18: Reminder: Quiz II in-class on Dec 5.

- 12/03/18: Lecture notes 19 posted.

- 11/28/18: Lecture notes 18 posted.

- 11/24/18: Lecture notes 17 and Homework 5 posted.

- 11/19/18: Lecture notes 16 posted.

- 11/14/18: Lecture notes 15 and Homework 4 posted.

- 11/13/18: Lecture notes 14 posted.

- 11/07/18: Lecture notes 13 posted.

- 11/04/18: Lecture notes 12 and Homework 3 posted.

- 10/29/18: Lecture notes 11 posted.

- 10/27/18: Zhisheng's office hours next week will be moved to Tue, Oct 30, 9:30–11:00am, in Jones 308.

- 10/27/18: Reminder: Quiz I in-class on Oct 31.

- 10/24/18: Lecture notes 10 and Homework 2 posted.

- 10/22/18: Lecture notes 9 posted.

- 10/19/18: Lecture notes 8 posted.

- 10/17/18: Lecture notes 7 posted.

- 10/15/18: Lecture notes 6 and Homework 1 posted.

- 10/12/18: Lecture notes 5 posted.

- 10/10/18: Lecture notes 4 posted.

- 10/09/18: Lek-Heng's office hours on Wed, Oct 10, will be in Jones 111 (not the usual location).

- 10/08/18: Lecture notes 3 posted.

- 10/06/18: Homework 0 posted.

- 10/03/18: Make-up lectures 3:00–4:20pm on Fri, Oct 12 and Fri, Oct 19, in Rosenwald 15.

- 10/03/18: Lecture notes 1 and 2 posted.

- 09/30/18: Check back regularly for announcements.

**Location:** Room 15, Rosenwald
Hall.

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

**Instructor:** Lek-Heng
Lim

Office: Jones 122C

`lekheng(at)galton.uchicago.edu`

Tel: (773) 702-4263

**Office hours:** Wed, 1:00–2:30pm, Jones 122C.

**Course Assistant I:** Greg
Naitzat

Office: Jones 203/204

`gregn(at)galton.uchicago.edu`

**Office hours:** Tue, 3:00–4:30pm, Jones 304.

**Course Assistant II:** Zhisheng
Xiao

Office: Jones 203/204

`zxiao(at)uchicago.edu`

**Office hours:** Thu, 3:30–5:00pm, Jones 308.

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 26, due: Dec 5)

- Problem Set 4 (posted: Nov 14, due: Nov 26)

- Problem Set 3 (posted: Nov 5, due: Nov 14)

- Problem Set 2 (posted: Oct 24, due: Nov 5)

- Problem Set 1 (posted: Oct 15, due: Oct 24)

- Problem Set 0 (posted: Oct 6, due: Oct 15)

**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, Fall 2017. 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 31, 3:00–4:20pm. Quiz II on
Wed, Dec 5, 3:00–4:20pm.

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.