Computational Linear Algebra

Introduction: Matrices, eigenvalues, norms, spectral radius, condition number, basic stability estimates. Basic Direct Methods: Computational techniques based on Gauss Elimination Method, error estimates, stability, pivot strategies, implementation, factorization LU, Cholesky, Doolittle-Crout, LDLT, QR algorithms.  Iterative Methods: General theory, methods (Jacobi, Gauss Seidel), relaxation techniques (JOR, SOR), general Richardson iteration, gradient methods, conjugate gradient method, introduction to Arnoldi, Krylov, GMRES methods. Computational Methods for Eigenvalues/Eigenvectors: Introduction to geometrical properties of eigenvalues, stability estimates, power method, QR method, Householder matrices, inverse power method. Nonlinear Systems: Introduction to general iterative methods, Newton-Raphson method, Quasi-Newton Methods, implementation issues.