Monash University Handbook

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. Students 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. Students will learn to implement the computational methods efficiently, and will learn how to thoroughly test their implementations for accuracy and performance. Students 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.

Outcomes

On completion of this unit students will be able to:

Explain 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.
Explain and apply notions of conditioning, stability, accuracy, convergence, convergence speed and computational cost.
Demonstrate proficiency in the main linear algebra algorithms for solving linear systems, least-squares problems, eigenvalue decompositions, and other matrix decompositions, and apply them to problems in science, engineering, technology and big data analytics.
Implement advanced computational linear algebra methods, and demonstrate the correctness and efficiency of the implementations in systematic computational tests.
Demonstrate advanced skills in the written and oral presentation of theoretical and applied computational linear algebra problems.

Assessment

End of semester examination (3 hours): 60% (Hurdle)

Continuous assessment: 40% (Hurdle)

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for both the end-of-semester examination and continuous assessment components.