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AMS 526, Numerical Analysis I
Direct and indirect methods for solving simultaneous linear equations and matrix inversion, conditioning, and round-off errors. Computation of eigenvalues and eigenvectors.
Co-requisites: AMS 510 and AMS 595
3 credits, ABCF grading
Required Textbooks for Fall 2022:
"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
1) Demonstrate mastery of concepts and numerical methods for solving systems of linear
* 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
* 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.