University of Chicago

Fall 2015

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.

- 12/02/15: Lecture Notes 17 posted and Lecture Notes 16 updated.

- 12/01/15: Homework 5 posted.

- 11/19/15: Quiz II next Tue, Nov 24, in class.

- 11/19/15: Lecture Notes 16 posted.

- 11/17/15: Lecture Notes 15 posted and Lecture Notes 14 updated.

- 11/12/15: Lecture Notes 14 and Homework 4 posted.

- 11/10/15: Lecture Notes 13 posted.

- 11/06/15: Lecture Notes 12 posted.

- 11/03/15: Lecture Notes 11 and Homework 3 posted.

- 10/27/15: Lecture Notes 10 posted.

- 10/23/15: Lecture Notes 9 posted.

- 10/20/15: Lecture Notes 8 posted.

- 10/19/15: Lecture Notes 7 and Homework 2 posted.

- 10/18/15: Make-up lecture Mon, Oct 19, from 5:30–8:30pm in Eckhart 133.

- 10/16/15: Lecture Notes 6 posted.

- 10/13/15: Lecture Notes 5 posted.

- 10/12/15: Lecture Notes 4 posted.

- 10/10/15: Marc will give a Matlab tutorial on Fri, Oct 16, 10:30–11:00am in Eckhart 131.

- 10/10/15: Lecture Notes 3 posted.

- 10/07/15: Lecture Notes 2 and Homework 1 posted.

- 10/05/15: Marc will hold office hours this week on Wed, Oct 7, 3:00–4:00pm in Eckhart 131.

- 09/29/15: No class on Tue, Oct 6. Make-up lectures on Mon, Oct 12 and Mon, Oct 19, from 5:30–8:30pm in Eckhart 133.

- 09/29/15: Quiz I on Thu, Oct 29, Quiz II on Tue, Nov 24 (both in-class).

- 09/29/15: Lecture Notes 1 and Homework 0 posted.

- 09/29/15: Check back regularly for announcements.

**Location:** Room 133, Eckhart
Hall.

**Times:** 3:00–4:20pm on Tue and Thu.

**Instructor:** Lek-Heng
Lim

Office: Eckhart 122

`lekheng(at)galton.uchicago.edu`

Tel: (773) 702-4263

**Office hours:** 1:30–3:30pm on Wed.

**Course Assistant:** Marc
Goessling

Office: Eckhart 131

`goessling(at)galton.uchicago.edu`

**Office hours:** 11:00am–12:00n on Fri.

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: Dec 01, due: Dec 9)

- Problem Set 4 (posted: Nov 12, due: Dec 01)

- Problem Set 3 (posted: Nov 03, due: Nov 17)

- Problem Set 2 (posted: Oct 20, due: Nov 06)

- Problem Set 1 (posted: Oct 08, due: Oct 20)

- Problem Set 0 (posted: Sep 29, due: Oct 08)

**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. Related course homepages from Fall 2005 and Spring 2006.

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

**Exam dates:** Quiz I on Oct 29. Quiz II on Nov 24.

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.