ECE452 Special Topics in Applied Mathematics

Introductory Review of Matrix Theory. (Linear Vector Spaces, Square Matrices, Eigenvalues and Eigenvectors, Norms.) Direct Methods of Solution of Linear Systems, GaussElimination Method and itsModifications, LU Factorization, Method ofCholesky, Elements of Perturbation Theory, Iterative Improvement of Numerical Solution of Linear Systems. Iterative Methods of Solution of Linear Systems, Classical Iterative Methods,General Iterative Method, Methods of Jacobi and Gauss-Seidel, Convergence Acceleration Techniques (Extrapolation), SOR Method, Symmetric SOR (SSOR) Method, Applications (Difference Equations, Tensor Products). Minimization Methods for the Solution of Linear Systems (Methods of Steepest Descent, Conjugate Directions,Conjugate Gradient (CG), Preconditioned CG). Least Squares Method, Theory of Linear Problem, Gram-Schmidt Method, QR Analysis, Householder and Givens Transformations, Analysis of Singular Values (SVD). Numerical Methods for the Determination of the Eigenvalues and Eigenvectors, Power Method and its Variations, Determination of other Eigenvalues.

Students will be able to

• Understand the basic concepts of Numerical Linear Algebra (E.g., Solution of linear systems of algebraic equations by using direct and iterative methods).
• Solve practical problems as those that are produced after the discretization Partial Differential Equations.
• Analyze the accuracy of the obtained solutions as well as the convergence and the stability of the algorithms that are used.
• Compare similar algorithms and select the one that is most appropriate for the characteristics of the particular problem considered