AMS 526, Numerical Linear Algebra
Introduction to key concepts and algorithms in numerical linear algebra and their
applications in computational and data sciences. Topics include direct and iterative
methods for solving simultaneous linear equations, least squares problems, computation
of eigenvalues and eigenvectors, and singular value decomposition.
Prerequisites: AMS 510 and AMS 595
3 credits, ABCF grading
Fall Semester
Required Textbooks for Spring 2026:
"Matrix Computations" by Gene H. Golub and Charles F. Van Loan, 4th Edition, John Hopkins University Press, 2013; ISBN: 978-1-421-40794-4
"Numerical Linear Algebra" by Lloyd N. Trefethen and David Bau, III; Society for Industrial and Applied Mathematics; 1997; ISBN: 978-0-898713-61-9
Learning Outcomes:
1) Demonstrate mastery of concepts and numerical methods for solving systems of linear
equations:
* Gaussian elimination and its variants;
* Cholesky and LDL' factorizations.
2) Demonstrate master of concepts of orthogonality and numerical methods for linear
least squares problems:
* Orthogonal matrices, projectors, and linear least squares;
* QR factorization using Gram-Schmidt orthogonalization, Householder reflectors,
and Givens rotation.
3) Understand and apply conditioning and stability:
* Norms, condition numbers, and effect of rounding errors;
* Stable and backward stable algorithms;
* Backward error analysis of fundamental algorithms.
4) Demonstrate mastery of concepts and analyses based on eigenvalues and singular
values and their numerical computations:
* Singular value decomposition and eigenvalue decomposition;
* Power method and similarity transformations;
* Reduction to Hessenberg and tridiagonal forms.
5) Understand and use iterative methods for solving large sparse linear systems and
computing eigenvalues:
* Conjugate gradient method, GMRES, and other Krylov subspace methods;
* Lanczos and Arnoldi iterations;
* Preconditioners for iterative methods.
6) Demonstrate programming skills for numerical methods using the abstractions of linear algebra.