Course Listings

2025 Spring > GRAD > MATH > MATH 626

Numerical Linear Algebra

Course #: MATH 626

Description:
This course introduces the essential ideas and computational techniques that modern scientists or engineers will need in order to carry out their work. In most scientific modeling projects, investigators have to deal with very large systems of linear equations, understanding of which requires powerful computers, and a firm understanding of the vast number of existing pertinent algorithms. The main goal of the course is to provide an introduction to algorithmic and mathematical foundations of high-performance matrix computations. Topics include linear algebraic systems, the singular value decomposition (SVD) of a matrix and some of its modern applications. We will discuss Principal Component Analysis (PCA) and its applications to data analysis. We will study linear transformations and change of basis. We will discuss complex vector spaces and the Jordan canonical form of Matrices. We will discuss non-negative matrices and Perron-Frobenius Theory. We will explain multiple matrix factorizations, such as LU, QR, NMF. For each of these topics we will discuss numerical computer algorithms and their implementations. In particular we will discuss in detail eigenvalue estimation, including iterative and direct methods, such as Hausholder methods, tri-diagonalization, power methods, and power methods with shifts. We will explain concepts of numerical analysis that are important to consider when we talk about the implementation of algorithms, such as stability and convergence. We will discuss iterative methods as well as direct ones, their advantages and disadvantages. The methods and their applications will be illustrated using a common programming language such as python and/or R. The course will emphasize mathematical and software engineering methods that will allow students to fully participate at all levels of algorithm design and implementation.
Pre Requisites:
Pre-requisite: MATH 625 or permission of instructor