Fundamental knowledge and understanding of methods in numerical mathematics. Qualifying of students for solving of problems in this area by using scientific acts and methods. Ability to follow contemporary achievements in the area of numerical mathematics and its applications, especially in technique and Engineering. Realization of numerical methods by using the program systems Matlab, Mathematica.

Upon successful completion of this course, students should be able to: • They are skilled in solving mathematical models, resulting in problem solving in science, technic and engineering sciences, by methods of numerical mathematics in approximation theory, numerical differentiation and integration, the theory of iterative processes, numerical linear algebra, numerical solution of differential equations. • Locate errors that occur in the process of calculation, following their spreading and apply the knowledge gained in the construction of stable numerical methods. • Manage with implementation of numerical methods in MATLAB programming system. • Monitor contemporary achievements in the field of numerical mathematics and its applications, particularly in technic and engineering sciences.

Elements of the errors theory. IEEE-754-2008. Classes singe and double in Matlab. Machine precision. Errors of approximate values of functions. Inverse problem for error. Condition of problem. Interpolation, Lagrange and Newton interpolation polynomials. Matlab function interp1. Numerical differentiation. Matlab function diff. Numerical methods for solving of nonlinear equations and systems. Interpolation quadrature formulas. Matlab functions integral, trapz. Methods of error estimation. Generalization to multiple integrals. Construction of Gauss formulas from Jacobi matrix by QR algorithm. Modification of Gauss formulas. Radau and Lobatto quadratures. Kronrod schemes. Gauss-Turán quadratures and generalizations. Convergence of quadrature processes. Formulas of trigonometric type. Integration of fast oscillating functions. Interpolating cubature formulas. Review of cubature formulas for some specific areas and certain weight functions. Numerical linear algebra. Gassuian elimination. LU factorization. Perturbation analysis. Iterative methods. Functions linsolve, lu in Matlab. Approximation theory. Bernstein theorem. Least squares approximation. Discrete least squares approximation. Chebyshev mini-max approximation. Implementation of linear and nonlinear regression in Matlab. ODE. Cauchy problem. Euler methos. Convergence analysis. Crank-Nickolson method. Zero stability. Stability on unbounded intervals. Higher order methods. Predictor-corrector methods. Systems of ODE. Runge-Kutta methods. ODE functions family in Matlab. PDE. Classification. Elliptic equations. Variational formulation of the Dirichet problem. Neuman problem. Finite difference method. Finite element method. Eigen value problem for elliptic operator. Parabolic equations. Variational formulation. Hyperbolic equations. Finite difference methods. Finite element methods. PDE toolbox in Matlab.

Elements of the errors theory. IEEE-754-2008. Classes singe and double in Matlab. Machine precision. Errors of approximate values of functions. Inverse problem for error. Condition of problem. Interpolation, Lagrange and Newton interpolation polynomials. Matlab function interp1. Numerical differentiation. Matlab function diff. Numerical methods for solving of nonlinear equations and systems. Interpolation quadrature formulas. Matlab functions integral, trapz. Methods of error estimation. Generalization to multiple integrals. Construction of Gauss formulas from Jacobi matrix by QR algorithm. Modification of Gauss formulas. Radau and Lobatto quadratures. Kronrod schemes. Gauss-Turán quadratures and generalizations. Convergence of quadrature processes. Formulas of trigonometric type. Integration of fast oscillating functions. Interpolating cubature formulas. Review of cubature formulas for some specific areas and certain weight functions. Numerical linear algebra. Gassuian elimination. LU factorization. Perturbation analysis. Iterative methods. Functions linsolve, lu in Matlab. Approximation theory. Bernstein theorem. Least squares approximation. Discrete least squares approximation. Chebyshev mini-max approximation. Implementation of linear and nonlinear regression in Matlab. ODE. Cauchy problem. Euler methos. Convergence analysis. Crank-Nickolson method. Zero stability. Stability on unbounded intervals. Higher order methods. Predictor-corrector methods. Systems of ODE. Runge-Kutta methods. ODE functions family in Matlab. PDE. Classification. Elliptic equations. Variational formulation of the Dirichet problem. Neuman problem. Finite difference method. Finite element method. Eigen value problem for elliptic operator. Parabolic equations. Variational formulation. Hyperbolic equations. Finite difference methods. Finite element methods. PDE toolbox in Matlab.

The course attendance conditions is determined by the curriculum of study program.

1. М.M. Spalević, M.S. Pranić, Numerical methods, Skver, Kragujevac, 2007 (http://mat.mas.bg.ac.rs) 2. G.V. Milovanović, М. Kovačević, М. Spalević, Numerical mathematics - Collection of solved problems, University of Niš, 2003 (http://mat.mas.bg.ac.rs) 3. G.V.Milovanović, Numerical analysis, Parts 1, 2, 3, Naučna knjiga, Beograd, 1991 4. B.S. Jovanović: Numerical methods for solving PDE, Math. Institute, Beograd 1989, pgs. 130 5. G. Mastroianni, G.V. Milovanović: Interpolation Processes - Basic Theory and Applications, Springer Monographs in Mathematics, Springer – Verlag, Berlin – Heidelberg, 2008, XIV+444 pp. 6. W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, 2004 7. W. Gautschi, Numerical Analysis: An Introduction, Birkhäuser, Boston, 1997 8. A. Quarteroni, F. Saleri, Scientific Computing with MATLAB, Springer, 2003. 9. S. Larsson, V. Thomee, Partial Differential with Numerical Methods, Springer, 2005 10. Software Matlab 11. Software Mathematica 12. A.S. Cvetković, М.М. Spalević, Numerical methods, University of Belgrade, 2013

Total assigned hours: 65

New material: 35
Elaboration and examples (recapitulation): 15

Auditory exercises: 0
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 0
Review and grading of seminar papers: 10
Review and grading of the project: 0
Test: 0
Test: 5
Final exam: 0

Activity during lectures: 10
Test/test: 0
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 60
Project: 0
Final exam: 30
Requirement for taking the exam (required number of points): 0

G. Mastroianni, G.V. Milovanović: Interpolation Processes - Basic Theory and Applications, Springer Monographs in Mathematics, Springer – Verlag, Berlin – Heidelberg, 2008. ISBN 9783540683469 (hbk.) ; W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, 2004. ISBN 0 19 850672 4; W. Gautschi, Numerical Analysis: An Introduction, Birkhäuser, Boston, 1997. ISBN 0-8176-3895-4 ISBN 0-7643-3895-4; A. Quarteroni, F. Saleri, Scientific Computing with MATLAB, Springer, 2003. ISBN_10 3-540-44363-0; S. Larsson, V. Thomee, Partial Differential with Numerical Methods, Springer, 2005. ISBN 978-3-540-88705-8