MMath&Phys Mathematics and Physics

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This module treats the main classes of problems in numerical linear algebra: linear systems, least square problems, and eigenvalue problems, covering both dense and sparse matrices. It provides analysis of the problems along with algorithms for their solution. It also uses MATLAB as tool for expressing and implementing algorithms and describes some of the key ideas used in developing high-performance linear algebra codes. Applications, such as machine learning and search engines, will be introduced throughout the module.

Students are not permitted to take, for credit, MATH46101 in an undergraduate programme and then MATH66101 in a postgraduate programme at the University of Manchester, as the courses are identical.

To develop understanding of modern methods of numerical linear algebra for solving linear systems, least squares problems and the eigenvalue problem.

On completion of the module, students will be able to

• construct some key matrix factorizations using elementary transformations,
• choose an appropriate numerical method to solve systems, least squares problems, and the eigenvalue problem.
• evaluate and compare the efficiency and numerical stability of different algorithms for solving linear systems, least squares problems, and the eigenvalue problem.
• design algorithms that exploit matrix structures such as triangularity, sparsity, banded structure, and symmetric positive definiteness,
• quantify the sensitivity of a linear system or least squares problem to perturbations in the data.

1. Basics. Summary/recap of basic concepts from linear algebra and numerical analysis: matrices, operation counts. Matrix multiplication, block matrices. [4]  

Matrix norms. Linear system sensitivity. [2]

2. Matrix factorizations. Cholesky factorization. QR factorization by Householder matrices and by Givens rotations. [3]  

LU factorization and Gaussian elimination; partial pivoting. Solving triangular systems by substitution. Solving full systems by factorization. Error analysis. Complete pivoting, rook pivoting. Numerical examples. [4]

3. Sparse and banded linear systems and iterative methods. Storage schemes for banded and sparse matrices.  LU Factorization, Markowitz pivoting. [2]    

Iterative methods: Jacobi, Gauss-Seidel and SOR iterations. Kronecker product. Krylov subspace methods, conjugate gradient method. Preconditioning. [4]

4. Linear least squares problem. Basic theory using singular value decomposition (SVD) and pseudoinverse. Perturbation theory. Numerical solution: normal equations. SVD and rank deficiency.  [3]

5. Eigenvalue problem. Basic theory, including perturbation results. Power method, inverse iteration. Similarity reduction. QR algorithm. [5]

30 lectures (two or three lectures per week), with a fortnightly examples class.

• Mid-semester coursework: 20%
• End of semester examination: weighting 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
 

Further Reccomended 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.
 

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

· You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently
· You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week
· There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading
· In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.