MATH639: Programming assignments
MATH639: Content of the final test, Spring 2004
Part I. Basic concepts in linear algebra:
vectors, matrices, norms, various normal forms of matrices,
spectrum of a matrix, eigenvalue bounds, Hermitian matrices
and their properties;
Part II. Basic iterative methods:
general iteration processes and their convergence, Richardson,
Jacobi, Gauss-Seidel, SOR methods and their convergence,
optimization of SOR, block versions, symmetric versions;
Part III. Acceleration of iteration methods:
Chebyshev acceleration, SOR acceleration, the concept of preconditioning,
various normal forms of the iteration methods;
Part IV. Variational methods:
method of steepest descent, convergence analysis, method
of conjugate directions as a basis for CG method, CG method,
derivation and convergence analysis, preconditioned CG;
Part V. Projection methods:
general framework for projection methods; one-dimensional
projection methods; MINRES - derivation; residual norm steepest descent;
GMRES - general idea, GMRES with Arnoldi orthogonalization,
Givens rotations and GMRES algorithm; preconditioned GMRES;
Part VI. Newton's method:
general setting, standard assumptions, local convergence,
modifications.
Last update 04/05/04