MATH46101/MATH66101 Numerical Linear Algebra
Announcements
- Good luck in the exam! Please read the Feedback section below.
- See this XKCD cartoon about machine learning.
- As we continue the section on iterative methods you might find it interesting to watch the video about Matrix-Based Network Computations by David Gleich. It's a nice introduction to matrices from networks and iterative methods such as Gauss-Seidel.
- You might like to take a look at these excellent videos on a course Mathematics 580: Introduction to Numerical Analysis I given at North Carolina State University by my colleague Professor Tim Kelley. There is quite a lot of material in common with our course.
- Numerical linear algebra is very much an "alive subject". Here are links to some NLA researchers mentioned in the course: Cleve Moler (The MathWorks), Tim Davis (Texas A&M University), Charlie Van Loan (Cornell University).
- Turing Award winner Professor John Hopcroft recently gave a very understandable introduction to deep learning, which I highly recommend. He explains the importance of matrices in the topic.
- All members of the class are very welcome to attend any of the weekly meetings of the Numerical Linear Algebra group, which take place on Tuesdays at 10am in Frank Adams 1. Feel free to ask one of the lecturers for more details. The group's "need to know" page has some information that you might find useful.
- See how this course fits into the Numerical Analysis Undergraduate Study Pathway.
Feedback
- Feedback on the last four years' exams can be found at the bottom of this page. Past papers for the last three years' exams can be found at this University web page.
Handouts
- Cholesky Factorization (updated October 2, 2017) ... Solutions to examples.
- QR Factorization ... Solutions to examples
- LU Factorization (updated October 20, 2017 to add missing reference) ... Solutions to examples.
- LU Factorization Example
- Banded and Sparse Linear Systems
- Iterative Methods ... Solutions to examples.
- The Kronecker Product
- CG Method and Preconditioning (added items to page 5 on October 23, 2017)
- The Linear Least Squares Problem (added Aside on generalized inverses on November 17, 2017) ... Solutions to examples.
Assignment
There is one assignment with deadline Friday November 17, 2017.- Assignment. M-file headers: qr_givens.m, qr_solve.m
Kahoot Quizzes
MATLAB
Background on MATLAB for those wishing to know more. MSc students: see also the introductory lectures you received about MATLAB.- Introduction to MATLAB
- MATLAB Functions
- MATLAB Cheatsheet
- You might also find this MATLAB tutorial website useful.
- MATLAB Worksheets
- Worksheet 1
- Worksheet 2. M-file template: funprobe.m
LaTeX
I encourage you to typeset the assignment in LaTeX. Some useful links:- Top Tips for New LaTeX Users Blog post version.
- How To Typeset an Ellipsis in a Mathematical Expression
Further Reading
- David Gleich, Expanders, Tropical Semi-Rings, and Nuclear Norms: Oh My!, XRDS: Crossroads, The ACM Magazine for Students, 19(3) 32-36, 2013. What does "The Matrix" have to do with "The Social Network"?
- Desmond J. Higham and Alan Taylor, The Sleekest Link Algorithm, Mathematics Today, 39(6):192-197, 2003. An article explaining the maths begind Google's PageRank algorithm.
- Nicholas J. Higham, Cholesky Factorization, WIREs Comp. Stat., 1(2):251-254, 2009.
- Nicholas J. Higham, Gaussian Elimination, WIREs Comp. Stat., 3(3):230-238, 2011.
- Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, In N. J. Higham, M. R. Dennis, P. Glendinning, P. A. Martin, F. Santosa, and J. Tanner, editors, The Princeton Companion to Applied Mathematics, pages 263-281. Princeton University Press, Princeton, NJ, USA, 2015.
- Nicholas J. Higham, The Singular Value Decomposition, In N. J. Higham, M. R. Dennis, P. Glendinning, P. A. Martin, F. Santosa, and J. Tanner, editors, The Princeton Companion to Applied Mathematics, pages 126-127. Princeton University Press, Princeton, NJ, USA, 2015.
- Gilbert Strang, Row Rank Equals Column Rank: Four Approaches, IMAGE (The Bulletin of the International Linear Algebra Society), 53:17, 2014.
Textbooks
The following list comprises a subset of the textbooks on the syllabus that are available electronically (e-books). Note that you must be either on campus or logged in via VPN in order to access SIAM e-books.
- Timothy A. Davis. Direct Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2006. ISBN 0-89871-613-6. xii+217 pp. E-book at SIAM
- James W. Demmel. Applied Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1997. ISBN 0-89871-389-7. xi+419 pp. E-book at SIAM.
- Desmond J. Higham and Nicholas J. Higham. MATLAB Guide, First edition, 2000. Second edition, 2005. Third edition, 2017. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2000. ISBN 0-89871-469-9. E-book at SIAM (second edition).
- Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, second edition, 2002. ISBN 0-89871-521-0. xxx+680 pp. E-book at SIAM.
- Yousef Saad. Iterative Methods for Sparse Linear System}. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, second edition, 2003. ISBN 0-89871-534-2. xviii+528 pp. E-book at SIAM.
- G. W. Stewart. Matrix Algorithms. Volume I: Basic Decompositions. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2001. ISBN 0-89871-503-2. xix+469 pp. E-book at SIAM.
- G. W. Stewart. Matrix Algorithms. Volume II: Eigensystems. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2001. ISBN 0-89871-503-2. xix+469 pp. E-book at SIAM.
Links Mentioned in Lectures
- The paper The QR Algorithm: 50 Years Later its Genesis by John Francis and Vera Kublanovskaya and Subsequent Developments, by Gene Golub and Frank Uhlig (2009).
- Cartoon (Daily Mirror, 1952) about Turing and Wilkinson.
- Online Appendix A: Introduction to Matrix Computations, from the book Numerical Methods in Scientific Computing, Volume I, G. Dahlquist and A. Björck, SIAM, 2008.
- The Origins of MATLAB (video), by Cleve Moler.
- The Golden Age of Scientific Computing (video), by Chris Johnson.
- @nhigham
- @walkingrandomly. Contains interesting maths and maths software information.
- @TeXtip - tips on (La)TeX.