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.

Offerings

S1-01-CLAYTON-ON-CAMPUS

Rules

Enrolment Rule

Contacts

Chief Examiner(s)

Dr Mark Flegg

Unit Coordinator(s)

Dr Mark Flegg

Notes

This Level 4 unit and its Level 3 counterpart MTH3320 share the same core content and learning activities such as seminars and applied classes. However, studies at Level 4 are distinguished from those at Level 3 by a deeper understanding of mathematical theories and their applications, higher levels of critical thinking, and greater autonomy in learning

Learning outcomes

On successful completion of this unit, you should be able to:

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.

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.

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.

Design, implement, and critically evaluate advanced computational linear algebra methods, demonstrating the correctness and efficiency of the implementations through systematic and rigorous computational tests.

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

Teaching approach

Active learning

Assessment

1 - Continuous assessment

2 - Final assessment - Exam (3 hours and 10 minutes)

Scheduled and non-scheduled teaching activities

Applied sessions

Seminars

Workload requirements

Workload

Learning resources

Ascher UM, Greif C. A First Course in Numerical Methods. Society for Industrial and Applied Mathematics; 2011. http://epubs.siam.org.ezproxy.lib.monash.edu.au/doi/book/10.1137/9780898719987 <http://epubs.siam.org.ezproxy.lib.monash.edu.au/doi/book/10.1137/9780898719987>

Bjorck A. Numerical methods in matrix computations. Springer; 2015. https://link-springer-com.ezproxy.lib.monash.edu.au/book/10.1007/978-3-319-05089-8 <https://link-springer-com.ezproxy.lib.monash.edu.au/book/10.1007/978-3-319-05089-8>

Demmel JW. Applied numerical linear algebra. Society for Industrial and Applied Mathematics; 1997. http://epubs.siam.org.ezproxy.lib.monash.edu.au/doi/book/10.1137/1.9781611971446 <http://epubs.siam.org.ezproxy.lib.monash.edu.au/doi/book/10.1137/1.9781611971446>

Gander W, Gander MJ, Kwok F. Scientific computing-An introduction using Maple and MATLAB. Springer; 2014. https://link-springer-com.ezproxy.lib.monash.edu.au/book/10.1007/978-3-319-04325-8 <https://link-springer-com.ezproxy.lib.monash.edu.au/book/10.1007/978-3-319-04325-8>

Linge S, Langtangen HP. Programming for Computations-MATLAB/Octave: A Gentle Introduction to Numerical Simulations with MATLAB/Octave. Springer; 2016. https://link-springer-com.ezproxy.lib.monash.edu.au/book/10.1007/978-3-319-32452-4 <https://link-springer-com.ezproxy.lib.monash.edu.au/book/10.1007/978-3-319-32452-4>

Quarteroni A, Sacco R, Saleri F. Numerical mathematics. Springer; 2010. https://link-springer-com.ezproxy.lib.monash.edu.au/book/10.1007/b98885 <https://link-springer-com.ezproxy.lib.monash.edu.au/book/10.1007/b98885>

Saad Y. Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics; 2003. http://www-users.cs.umn.edu/~saad/IterMethBook_2ndEd.pdf <http://www-users.cs.umn.edu/~saad/IterMethBook_2ndEd.pdf>

Saad Y. Numerical Methods for Large Eigenvalue Problems: Revised Edition. Society for Industrial and Applied Mathematics; 2001. http://www-users.cs.umn.edu/~saad/eig_book_2ndEd.pdf <http://www-users.cs.umn.edu/~saad/eig_book_2ndEd.pdf>

Trefethen LN and Bau D. Numerical linear algebra. Society for Industrial and Applied Mathematics; 1997. On overnight reserve in library.

Other unit costs

Costs are indicative and subject to change. Miscellaneous items required (unit course reader, printing, stationery) - $120

Availability in areas of study

Applied mathematics
Financial and insurance mathematics
Mathematical statistics
Mathematics
Pure mathematics