Selected topics in numerical analysis 1000-135AN

* Matrix eigenvalue problem. Conditioning of eigenvalues and eigenvectors. Power method, inverse iteration and Rayleigh quotient iteration (RQI). QR method. Convergence in the symmetric case. Orthogonal transformations. Jacobi and "divide and conquer" methods. Computational cost and numerical properties.

* SVD decomposition and irregular least squares problem.

* Iterative solution of large sparse systems of linear equations. CG and GMRES methods, their convergence and implementation. Examples of stationary iterations and their convergence condition. Short survey on other iterative methods (CGT, PCR, BiCG, multigrid, etc). Issues of parallel implementation and efficiency. Preconditioning and spectral equivalence.

* Iterative solution of systems of nonlinear equations. Banach's fixed point iteration. Newton's method and its variants. Convergence of these methods. Information on Kantorowich theorem. Stopping criteria. Information on continuation methods.

* Numerical quadrature in many dimensions. One dimensional quadratures. Tensor product quadratures in low dimension. Curse of dimensionality. Mone Carlo quadratures. Information on variance reduction and QMC.