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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 quadrature. 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). REQUIRED READING 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. ADDITIONAL READING S. C. Brenner, L. R. Scott: The mathematical theory of finite element methods, Springer 2007. W. Cheney, D. Kincaid: Numerical Mathematics and Computing, Fourth Edition, Brooks/Cole, Pacific Grove, 1999. P. G. Ciarlet: Introduction to numerical linear algebra and optimisation, Cambridge U. Press, Cambridge 1989. N. J. Higham: Accuracy and Stability of Numerical Algorithms, Second Edition, SIAM, Philadelphia 2002. R. M. M. Mattheij, S. W. Rienstra, J. H. M. ten Thije Boonkkamp: Partial Differential Equations: Modeling, Analysis, Computation, SIAM, Philadelphia, 2005. A. Ralston, P. Rabinowitz: A First Course in Numerical Analysis, Second Edition, McGraw-Hill, Singapore, 1978. M. Schatzman: Numerical Analysis, A Mathematical Introduction, Oxford University Press, Oxford, 2002.