University of Chicago

Fall 2012

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.

- 12/05/12: There will be a lecture on Dec 6.
- 12/02/12: Office hours this week moved to Mon, 1:00–2:30pm.
- 12/01/12: Homework 5 and Lecture notes 16 posted.
- 11/28/12: Lecture notes 15 posted.
- 11/27/12: Lecture notes 12, 13, 14 posted.
- 11/15/12: Homework 4 and Lecture notes 11 posted.
- 11/06/12: Lecture notes 10 posted.
- 11/01/12: Homework 3 and Lecture notes 9 posted.
- 10/28/12: Lecture notes 8 posted.
- 10/25/12: Lecture notes 7 posted.
- 10/24/12: Homework 2 and Lecture notes 6 posted.
- 10/23/12: Lecture notes 5 posted.
- 10/18/12: As announced in class, lecture notes and other references are kept in a non-public folder, please write to me for instructions on accessing them.
- 10/18/12: Lecture notes 3 & 4 posted.
- 10/14/12: This Mon (10/15) office hours moved to 1–3pm.
- 10/09/12: Lecture notes 2 posted.
- 10/07/12: Homework 1 posted.
- 10/04/12: Lecture notes 1 posted.
- 10/02/12: Check back regularly for announcements.

**Location:** Room 277, Ryerson
Physical Laboratory.

**Times:** 1:30–2:50pm on Tue/Thu.

**Instructor:** Lek-Heng
Lim

Office: Eckhart 122

`lekheng(at)galton.uchicago.edu`

Tel: (773) 702-4263

**Office hours:** 2:00–4:00pm, Mon

**Course Assistant:** Yunda
Zhong

Office: Eckhart 8

`ydzhong(at)galton.uchicago.edu`

**Office hours:** 5:30–6:30pm, Fri

The last three 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.

Except for the take-home final, collaborations are permitted but you will need to write up your own solutions. The problem sets are designed to get progressively more difficult. You will get at least seven days for each problem set.

- Problem Set 1 (posted: Oct 07; due: Oct 16)

- Problem Set 2 (posted: Oct 24; due: Nov 01)

- Problem Set 3 (posted: Nov 01; due: Nov 13)

- Problem Set 4 (posted: Nov 15; due: Nov 29)

- Problem Set 5 (posted: Dec 01; due: Dec 11)

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

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

**Grade composition:** 75% Problem Sets, 25% Final

We will use the 4th edition of Golub-Van Loan. Copies of selected sections will be distributed in class. You are all encouraged to buy a copy when the book is published in December (you are welcomed to browse a copy of this new edition during office hours).

- 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.