Advanced Numerical Analysis II

This course covers basic numerical methods in the graduate level. The main themes are numerical solutions of PDEs and numerical methods for eigenvalue problems. For numerical solutions of PDEs, some basic theories of finite difference methods for hyperbolic equations are discussed. The topics include stability, order of accuracy, consistency, Lax-equivalence theorem, multistep methods, dissipation and dispersion, system of equations in higher dimensions, splitting schemes. For numerical methods for eigenvalue problems, the topics include the Gershgorin theorem, power iteration, inverse power iteration, Householder and Givens matrices, QR factorization, QR algorithm, singular value decomposition, principle component analysis, rank deficient problems, regularization techniques.