MTH4320 · Computational linear algebra

Overview

The overall aim of this unit is to study the numerical methods for matrix computations that lie at the core of a wide variety of large-scale computations and innovations in the sciences, engineering, technology and data science. You will receive an introduction to the mathematical theory of numerical methods for linear algebra (with derivations of the methods and some proofs). This will broadly include methods for solving linear systems of equations, least-squares problems, eigenvalue problems, and other matrix decompositions. Special attention will be paid to conditioning and stability, dense versus sparse problems, and direct versus iterative solution techniques. You will learn to implement the computational methods efficiently, and will learn how to thoroughly test their implementations for accuracy and performance. You will work on realistic matrix models for applications in a variety of fields. Applications may include, for example: computation of electrostatic potentials and heat conduction problems; eigenvalue problems for electronic structure calculation; ranking algorithms for webpages; algorithms for movie recommendation, classification of handwritten digits, and document clustering; and principal component analysis in data science.

Areas of study: Applied mathematics Financial and insurance mathematics Mathematical statistics Mathematics Pure mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

1. Critically evaluate and synthesise the mathematical theory behind a selection of important numerical methods for linear algebra, including the derivation of the methods and the analysis of their properties.
2. Analyse and apply advanced concepts of conditioning, stability, accuracy, convergence, convergence speed, and computational cost, demonstrating a thorough understanding of these notions in complex scenarios.
3. Exhibit mastery in the most important linear algebra algorithms for solving linear systems, least-squares problems, eigenvalue decompositions, and other matrix decompositions, applying these methods to complex problems in science, engineering, technology, and big data analytics.
4. Design, implement, and critically evaluate advanced computational linear algebra methods, demonstrating the correctness and efficiency of the implementations through systematic and rigorous computational tests.
5. Communicate complex theoretical and applied computational linear algebra problems with clarity and precision, both in written and oral forms, suitable for academic and professional audiences

Workload

Three 1-hour seminars; One 2-hour applied class (in weeks 2-12) and 7 hours of independent study per week

Student experience, not official rules. Nothing here changes what the Handbook says.