## Numerical methods

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Course Unit Code Number of ECTS Credits Allocated Type of Course Unit * Level of Course Unit * Year of Study * 714-0087/01 5 ECTS credits Choice-compulsory Second Cycle First Year Summer Semester Face-to-face Czech Course succeeds to compulsory courses of previous semester CER365 doc. Ing. Martin Čermák, Ph.D. Basic problems of the numerical mathematics, errors in computations. Solving of equation f(x)=0: bisection method, regula-falsi, iterative method, Newton´s iteration, roots of polynomials. Numerical solution of systems of linear algebraic equations: LU-factorization, iterative methods, condition number of matrix, ill-conditioned matrices. Numerical solution of systems of nonlinear equations: Fixed-point iteration, Newton’s method. Interpolation and approximation of functions: Polynomial interpolation, interpolation by cubic spline functions, least squares approximation. Numerical integration: Trapezoid rule, Simpson’s rule, Richardson extrapolation, Monte Carlo method. The aim of this course is to acquaint students with the numerical solution of mathematical problems that arise in the other courses of their study and in the technical practice. The main accent lays in explanations of fundamental principles of numerical methods with emphases their general properties. It should lead to the ability in concrete situations to decide whether a numerical procedure is a suitable tool for solving a particular problem. An important ingredient of the course consists in the algorithmic implementation and in the utilization of existing computer programs specialized for numerical computations. The graduate of this course should know: - to recognize problems suitable for solving by numerical procedures and to find an appropriate numerical method; - to decide whether the computed solution is sufficiently accurate and, in case of need, to assess reasons of inaccuracies; - to propose an algorithmic procedure for solving the problem and to choice a suitable computer environment for its realization. 1. Course contents, the issue of errors, stability of calculations. 2. Solution of nonlinear equations, separation of roots, bisection method, regula-falsi method.. 3. Newton's method and fixed-point iterations. 4. Direct methods for solving linear equations, Gaussian elimination and LU-decomposition. 5. Eigenvalues and eigenvectors, numerical calculation. 6. Iterative methods for solving linear equations. 7. Iterative methods for solving nonlinear equations. 8. Interpolation by polynomials. 9. Interpolation by splines. 10. Least squares approximation. 11. Numerical differentiation and integration, Newton-Cotes formulae. 12. Extrapolation in the calculation of integrals. Gaussian integration formulas. 13. Numerical integration by Monte-Carlo method. 14. Reserve.  Forsythe, G., E., Malcolm, M.,A., Moler, B., C.: Computer Methods for Mathematical Computations. Prentice –Hall, Inc., Englewood Clifs, N.J. 07632 1977.  Buchanan, J., L., Turner, P., R.: Numerical Method and Analysis. McGraw-Hill, Inc., New York 1992. ISBN 0-07-112922-7  Boháč,Z., Častová,N.: Základní numerické metody. Skriptum VŠB-TUO, Ostrava 1997. ISBN 80-7078-975-1  Demidovič,B.,P.,Maron,J.,A.: Základy numerické matematiky. SNTL, Praha 1966.  Ralston,A.: Základy numerické matematiky. Academia, Praha, 1978.  http://homel.vsb.cz/~kuc14/teach_NM.html  Stoer, J., Burlish, R.: Introduction to Numerical Analysis. Springer-Verlag, New York, Berlin, Heidelberg 1992. ISBN 0-387-97878-X  Vitásek,E.: Numerické metody.SNTL, Praha 1987 Lectures, Individual consultations, Tutorials, Other activities Exercises evaluation and Examination Credit and Examination 100 (100) 51 Exercises evaluation Credit 20 (20) 10 Jiný typ úlohy Other task type 20 10 Examination Examination 80 (80) 30 Písemná zkouška Written examination 60 25 Ústní zkouška Oral examination 20 5