Numerical Linear Algebra

MATH66101 Numerical Linear Algebra 15 Semester 1 Postgraduate Taught Level 6 Francoise Tisseur Unit coordinator Department of Mathematics 1 FHEQ level (Framework for Higher Education Qualifications) ' Masters/Integrated Masters P4 ' European Credit Transfer & Accumulation System Rating : 7.5 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 introduces MATLAB as tool for expressing and implementing algorithms and describes some of the key ideas used in developing high-performance linear algebra codes (blocking, BLAS). Applications will be introduced throughout the module. 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 introduces MATLAB as tool for expressing and implementing algorithms and describes some of the key ideas used in developing high-performance linear algebra codes (blocking, BLAS). Applications will be introduced throughout the module. 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 <ul> <li> construct some key matrix factorizations using elementary transformations,</li> <li> choose an appropriate numerical method to solve systems, least squares problems, and the eigenvalue problem.</li> <li> evaluate and compare the efficiency and numerical stability of different algorithms for solving linear systems, least squares problems, and the eigenvalue problem.</li> <li> design algorithms that exploit matrix structures such as triangularity, sparsity, banded structure, and symmetric positive definiteness,</li> <li> quantify the sensitivity of a linear system or least squares problem to perturbations in the data.</li> </ul> <ul><li>Introduction. Summary/recap of basic concepts from linear algebra and numerical analysis: matrices, operation counts. [1 lecture]. Introduction to MATLAB. [2]. Matrix norms. Linear system sensitivity. [2]</li><li>Matrix factorizations. Cholesky factorization. QR factorization by Householder matrices and by Givens rotations. [5]. LU factorization and Gaussian elimination; partial pivoting. Error analysis. [2]. Block algorithms and their suitability for modern machine architectures. [1]. The BLAS and LAPACK. [1]</li><li>Linear systems. Solving triangular systems by substitution. Solving full systems by factorization. Application: Newton's method for nonlinear systems. [1]</li><li>Sparse and banded linear systems and iterative methods. LU factorization for banded and sparse matrices. Storage schemes. [1]. Iterative methods: Jacobi, Gauss-Seidel and SOR iterations. Krylov subspace methods, conjugate gradient method. Preconditioning. Application: differential equations. [4]</li><li>Linear least squares problem. Basic theory using singular value decomposition (SVD) and pseudoinverse. Perturbation theory. Numerical solution: normal equations. SVD and rank deficiency. Application: image deblurring. [5]</li><li>Eigenvalue problem. Basic theory, including perturbation results. Power method, inverse iteration. Similarity reduction. QR algorithm. Application: Google PageRank. [5]</li></ul> 0 Other 20% 1 Written exam 80% <ul> <li> Mid-semester coursework: 20%</li> <li> End of semester examination: weighting 80%</li></ul> 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.   MATH11022 Linear Algebra Pre-Requisite Compulsory 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. N Further Reccomended Reading <ul> <li> 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"?</li> <li> 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.</li> <li> Nicholas J. Higham, Cholesky Factorization, WIREs Comp. Stat., 1(2):251-254, 2009.</li> <li> Nicholas J. Higham, Gaussian Elimination, WIREs Comp. Stat., 3(3):230-238, 2011.</li> <li> 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.</li> <li> 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.</li> <li> Gilbert Strang, Row Rank Equals Column Rank: Four Approaches, IMAGE (The Bulletin of the International Linear Algebra Society), 53:17, 2014.</li> </ul> Lectures 11 Tutorials 11 0 128