Faculty of Science

Objectives

To introduce the basic concepts of numerical analysis and presents solution methods for linear and nonlinear equations.

Contents

Background in linear algebra (Vector spaces, matrices, matrix norms, special matrices, eigenvalues and eigenvectors, similarity transformations, nonnegative matrices, M-matrices); Background in numerical analysis (well-posed problems, condition numbers, stability of numerical algorithms, stability and convergence, a priori and a posteriori analysis, sources of errors in numerical algorithms, computer representation of numbers); Direct solution methods for linear systems (Stability analysis of linear systems, triangular systems, Gauss elimination, LU-factorization, Cholesky-factorization, QR-factorization, block systems); Iterative solution methods for linear systems (convergence analysis of iterative methods, Jacobi-, Gauss-Seidel- and relaxation methods, convergence, preconditioning, gradient and conjugate gradient methods); Approximation of eigenvalues and eigenvectors (vector iteration, QR-iteration, transformation methods); Determination of zeros of nonlinear systems (geometric approach: bisection method, secant-, regula falsi- and Newton’s methods; fixed point iterations, Horner scheme and Newton-Horner scheme).