Numerical Linear Algebra and Applications

Linear algebra is a fundamental technique that underlies many mathematical disciplines. Numerical linear algebra, in turn, is the basis for scientific computation. This course covers the basic techniques, analysis methods, and details of implementing numerical linear algebra and computing software applications. The course emphasises the strong link between theoretical concepts, algorithm formulation, and practical implementation, which makes numerical linear algebra widely applicable in various fields. The recent advancements in data science heavily rely on extending concepts from numerical linear algebra, which will also be discussed in the course. This course is designed for MSc students with a background in mathematics. It covers the fundamentals of matrix algebra, computer arithmetic, conditioning and condition numbers, stability of numerical algorithms, vector and matrix norms, convergent matrices, stability of non-linear systems, sensitivity analysis, singular value decomposition (SVD), algebraic and geometric properties of SVD, least square solutions, Householder matrices and applications, the QR method, the Power method and applications, and the Jacobi method for finding the eigenvalues of a given matrix. The course has wide-ranging applications in fields such as data science, including machine learning, artificial intelligence, data visualization, and business intelligence.