**Lectures:** TR 1:30-2:45 pm

**Location:** Clough 423

**Instructor:** Edmond Chow

**E-mail:**

**Office Hours:** After class and by appointment

**TA:** Luke Erlandson

**TA E-mail:** sudo@gatech.edu

**TA Office Hours:** Tuesdays 4:30-5:30 at KACB 1324; Fridays 3:45-4:45 at CODA S1377K

**Course Description**

Introduction to the state-of-the-art iterative methods for solving linear and nonlinear systems of equations. This will be a very practical course, involving Matlab programming and a project.

**Prerequisites**

**Topics**

- Sparse matrices and review of direct methods
- Basic iterative methods (splitting methods, Jacobi, Gauss-Seidel, SOR)
- Chebyshev iterative method and matrix polynomials
- Krylov subspace methods (conjugate gradient method, GMRES, etc.)
- Projection method framework
- Related ideas for large-scale eigenvalue problems
- Methods based on biorthogonalization (if there is time)
- Iterative methods for linear least squares
- Preconditioning
- Multigrid methods
- Domain decomposition (if there is time)
- Nonlinear systems of equations (fixed point methods, Newton, Broyden, Newton-Krylov and other Newton variants for large problems)
- Line search and global convergence
- Contraction mapping and local convergence theory
- Nonlinear least squares (Gauss-Newton, Levenberg-Marquardt)
- Related ideas in optimization, e.g., stochastic gradient descent

**Learning Objectives**

Students will develop facility with iterative methods for the numerical solution of linear and nonlinear systems, and their analysis. The students will be able to:

- Given a linear or nonlinear system, choose an appropriate numerical solution method based on the properties of the system
- Evaulate a method for its convergence and computational cost, including parallel computing aspects
- Diagnose convergence problems of iterative solution methods
- Select or design a method or approach for preconditioning the solution of specific problems
- Use Matlab or other numerical software for solving systems of equations

**Grading**

50% Assignments and exercises.

40% Project (project options will be given in class).

10% Class participation.

**Audit and Pass/Fail**

Please inform the instructor and TA if you are taking the course for audit or pass/fail. If you wish to take the course for audit credit, you do not need to do a project, but you must attempt all the assignments and score at least 20% on each of them. If you wish to take the course as pass/fail, you need to score 50% or higher to pass the course. To accomplish this, you will likely need to make some attempt at the project.

**Recommended Texts**

__Iterative Methods for Sparse Linear Systems__, 2nd edition, by Yousef Saad, SIAM, 2003.__Numerical Methods for Unconstrained Optimization and Nonlinear Equations__, by J. E. Dennis, Jr. and Robert B. Schabel, SIAM, 1996.__Matrix Computations__, by Gene Golub and C. F. van Loan. Any edition is fine.