Numerical analysis The goal of the course Except as regular

```NUMERICAL ANALYSIS
THE GOAL OF THE COURSE
Except as regular introductory course in numerical analysis, this course is designed to cover the greater part of
the contents of the access exam in Numerical analysis as part of the doctoral studies in mathematics.
After successful examination of this course, students will be able to:

demonstrate knowledge and understanding of the concepts and results of numerical mathematics, solve
basic numerical problems analytically and using advanced computation on computers

independently use mathematical reading of numerical analysis in Croatian and English language
THE SYLLABUS
Errors and numerical stability
Sources of errors, error propagation. Computer arithmetic and round off errors. The stability of problems and
algorithms. Errors in summations. Computing the value of a polynomial and some elementary functions at the
given point.
Systems of linear equations
Direct methods, Gaussian elimination, LU factorization. Definite systems and the Cholesky factorization. Condition of the linear system, perturbation theory for linear systems and algorithms.
Iterative methods for linear systems
General theory of iterative methods. The classical iterative methods and their convergence: Jacobi, Gauss-Seidel,
JOR and SOR methods. The optimal choice of the relaxation parameter for special systems. Iterations from the
Krylov subspaces and their convergence. Arnoldy and GMRES methods.
Matrix formulation of linear least squares problem
Solve least squares problems using the QR factorization and singular value decomposition (SVD). The MoorePenrose inverse.
The eigenvalue problem
General theory of symmetric and nonsymmetric eigenvalue problem. Variational characterization of eigenvalues
and perturbation theory of Hermitian matrices. Reduction to Hessenberg and tridiagonal form. General QR
algorithm. The algorithms for symmetric matrices, classical Jacobi method, QR iteration, modern methods
(review). Algorithms for computing the singular value decomposition. Application of SVD.
Interpolation
Interpolation polynomials, estimation of error, convergence, optimal selection of nodes. Hermite interpolation.
Numerical derivation. Piecewise polynomial interpolation, linear and cubic spline, B-splines. Trigonometric and
rational interpolation, continued fractions.
Aproximation
Approximation in various norms. Discrete and continuous least squares problem. Orthogonal systems of
functions. Orthogonal polynomials (continuous and discrete). Generalized Horner scheme. Continuous and
discrete Fourier approximation. Min-max and almost min-max approximation. Smoothing the data.
Numerical integration
Newton-Cotes integration formulas. The trapezoidal and Simpson's rule. The composite formulas. The
trapezoidal rule for periodic functions,.The Euler-MacLaurin formula, the Romberg algorithm. The Gaussian
Rootfinding for nonlinear equations
The bisection method, simple function iteration, Newton's method. A general theory for one-point iteration
methods, roots of polynomials. Newton's method for nonlinear systems. The secant method.
Numerical solution of ordinary differential equations
Initial problems. Single-step methods, explicit and implicit Euler method, Runge-Kutta methods, implicit
trapezoidal method. Multistep methods, predictor-corrector couples. General theory of linear difference schemes,
consistency, stability, convergence (brief overview).
Numerical solutions of partial differential equations
Boundary problems, discretization using finite differences. Some practical problems and frequent equation in
practice. Finite differences in the time and spatial variables. Parabolic equations, explicit, implicit and CrankNicolson method. Elliptic problems, Laplace and Poisson equation on a rectangle, difference schemes.
Comparison of iterative methods for discrete Poisson equations on the square. Multigrid methods (a brief
overview).
K. E. Atkinson: An Introduction to Numerical Analysis, Third Edition, John Wiley & Sons, New York, 1988.
J. W. Demmel: Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
W. Gautschi: Numerical Analysis, An Introduction, Birkhauser, Boston, 1997.
G. Golub and C. F. Van Loan: Matrix computations. The John Hopkins University Press, Baltimore 2012.
J. Stoer, R. Bulirsch: Introduction to Numerical Analysis, Second Edition, Texts in Applied Mathematics
12, Springer Verlag, New York, 1992.