University of Chicago

Fall 2020

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.

- 09/22/20: This year's class will be conducted entirely on Canvas. This webpage would be used only as a mirror repository for course materials.

**Location:** Lectures held online through Canvas.

**Times:** 4:10–5:30pm on Mon and Wed.

**Instructor:** Lek-Heng
Lim

Office: Jones 122C

`lekheng(at)uchicago.edu`

Tel: (773) 702-4263

**Office hours:** Tue 2:00–4:00 pm.

**Course Assistant I:** Zhen Dai

Office: Jones 307

`zhen9(at)uchicago.edu`

**Office hours:** Wed 7:00–9:00 pm.

**Course Assistant II:** Zehua Lai

Office: Jones 307

`laizehua(at)uchicago.edu`

**Office hours:** Thu 2:00–4:00 pm.

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 4 (posted: Nov 19, due: Dec 4)

- Problem Set 3 (posted: Nov 5, due: Nov 18)

- Problem Set 2 (posted: Oct 24, due: Nov 4)

- Problem Set 1 (posted: Oct 12, due: Oct 23)

- Problem Set 0 (posted: Sep 30, due: Oct 11)

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

- Course homepage from Fall 2010, Fall 2011, Fall 2012, Fall 2013, Fall 2014, Fall 2015, Fall 2016, Fall 2017, Fall 2018, Fall 2019.

**Grade composition:** Five problem sets + one two-hour take-home
exam (open book, open notes), with lowest score of the six dropped.

**Exam date:** Wed, Dec 9, 1:30–3:30pm CST.

We will not use any specific book but the following are all useful references.

- 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.