This course covers both basic and state-of-the-art concepts, algorithms, theories, and implementations in numerical linear algebra. Students will learn and practice the subjects from the viewpoints of application, mathematics, and computing. We plan to cover the following topics.

* Fundamentals of scientific computing
   - Floating point arithmetic
   - Perturbation theory
   - Condition number
   - Numerical stability
   - Roundoff error analysis
   - Memory architecture and parallel computer
   - Computation and communication

* Linear systems: direct methods
   - LU decomposition
   - Error analysis (Pivoting and condition number)
   - Blocking algorithms for higher performance
   - Cholesky decomposition for symmetric positive definite matrices
   - Factorization for sparse matrices with reordering

* Least squares problems
   - Normal equations
   - QR decomposition
   - Singular value decomposition (SVD)
   - Orthogonal transformations (Householder, Givens rotations, others)

* Linear systems: iterative methods
   - Stationary iterative methods (Jacobi, Gauss-Seidel, SOR, SSOR)
   - Projection operators
   - Fundamentals of project methods
   - Arnoldo's method
   - Conjugate method (CG)
   - Generalized Minimal Residual (GMRES)
   - Other Krylov methods (BiCG, QMR, CGS, Bi-CGSTAB)
   - Preconditioning techniques

* Eigenvalue problems
   - Rayleigh quotient based methods