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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-prerequisite:  AMS 505 
3 credits, ABCF grading 

Fall Semester

Required Textbooks:

"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

AMS 526 Instructor Page

 

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.

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