Lecturer: Randa Herzallah

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 lectures

Assessment: Written exam 85%, Assignments 15%

Formal registration prerequisites: None

Assumed knowledge: The following core first and second year modules are particularly important:

• MA151 Algebra 1, MA149 Linear Algebra, MA148 Vectors and Matrices - providing methodological foundations
• MA124 Mathematics by Computer - being a useful introduction to some of the scientific computing aspects and programming elements of the module
• MA251 Algebra 1: Advanced Linear Algebra - developing more complex techniques in matrix classification and their properties in particular mathematical settings of interest
• MA259 Multivariable Calculus - providing the language and concepts needed for some of the typical proofs we will encounter.

Useful background: Some knowledge of numerical concepts such as accuracy, iteration and stability as provided in MA261 Differential Equations: Modelling and Numerics will become important in the context of this module. General interdisciplinary curiosity will also be supported through interactions with areas such as medical imaging and data science.

Synergies: The following modules link up well with Matrix Analysis and Algorithms, either through methodology, computational or application-oriented content:

• MA3K1 Mathematics of Machine Learning
• MA3H0 Numerical Analysis and PDEs
• MA4G7 Computational Linear Algebra and Optimisation
• PX390 Scientific Computing
• PX425 High Performance Computing in Physics

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA4M4 Topics in Complexity Science
• MA4J5 Structures of Complex Systems

Content: Many large scale problems arising in data analysis and scientific computing require to solve systems of linear equations, least-squares problems, and eigenvalue problems, for which highly efficient solvers are required. The module will be based around understanding the mathematical principles underlying the design and the analysis of effective methods and algorithms.

Aims: Understanding how to construct algorithms for solving some problems central in numerical linear algebra and to analyse them with respect to accuracy and computational cost.

Objectives: At the end of the module you will familiar with concepts and ideas related to:

• Various matrix factorisations as the theoretical basis for algorithms
• Assessing algorithms with respect to computational cost
• Conditioning of problems and stability of algorithms
• Direct versus iterative methods.

Books:

AM Stuart and J Voss, Matrix Analysis and Algorithms, script.

G Golub and C van Loan, Matrix Computations, 3. ed., Johns Hopkins Univ. Press, London 1996.

NJ Higham, Accuracy and Stability of Numerical Algorithms, SIAM 1996.

RA Horn and CR Johnson, Matrix Analysis, Cambridge University Press 1985.

D Kincaid and W Cheney, Numerical Analysis, 3. ed., AMS 2002.

LN Trefethen and D Bau, Numerical Linear Algebra, SIAM 1997.

Additional Resources