Numerical methods

Thorough knowledge and understanding of numerical mathematics methods. Training students to solve problems in this field using scientific procedures and methods. Ability to follow modern achievements in the field of numerical mathematics and its applications, especially in technology and engineering sciences. Implementation of numerical methods using the Matlab and Matematica programming systems.

Upon successful completion of this course, students should be able to: • Understand the solution of mathematical models that arise when solving problems in science, technology and engineering, numerical mathematics methods in the theory of approximations, numerical differentiation and integration, the theory of iterative processes, numerical linear algebra, numerical solution of differential equations. • Locate errors that occur in the calculation process, monitor their spread and apply the acquired knowledge in the construction of numerically stable procedures. • be proficient in the implementation of numerical methods in the MATLAB programming system. • keep up with modern developments in the field of numerical mathematics and its applications, especially in technology and engineering sciences

Elements of error theory. IEEE-754-2008. Classes single and double in Matlab. Machine accuracy. Errors of approximate values ​​of functions. Inverse error problem. Conditionality of the problem. Interpolation, Lagrange and Newton interpolation polynomials. Matlab function interp1. Numerical differentiation. Matlab function diff. Numerical methods for solving nonlinear equations and systems. Quadrature formulas of interpolation type. Matlab functions integral, trapz. Methods for estimating the residue. Generalization to multiple integrals. Construction of Gaussian formulas from the Jacobian matrix using the QR algorithm. Modifications of Gaussian formulas. Radau and Lobato type formulas. Kronrod schemes. Gauss-Turan quadratures and generalizations. Convergence of quadrature processes. Trigonometric-type formulas. Integration of fast oscillatory functions. Interpolation cubic formulas. Review of cubic formulas for some special areas and certain weight functions. Numerical linear algebra. Gaussian elimination. LU factorization. Conditionality of systems of linear equations. Iterative methods. Functions linsolve, lu in Matlab. Approximation theory. Bernstein's theorem. Mean-square approximation. Discrete mean-square approximation. Chebyshev's mini-max approximation. Implementation of linear and nonlinear regression in Matlab. ODE. Cauchy problem. Euler's method. Convergence of methods. Crank-Nicholson method. Stability of methods. Stability on unbounded intervals. Higher-order methods. Predictor-corrector methods. ODE systems. Runge-Kut methods. ODE family implementation in Matlab. PDJ. PDJ classification. Elliptic equations. Variational formulation of the Dirichlet problem. Neumann problem. Finite difference method for elliptic equations. Finite element method for elliptic equations. Eigenvalue problem for elliptic equations. Parabolic equations. Variational formulation. Hyperbolic equations. Finite difference methods. Finite element methods. PDE toolbox in Matlab.

Elements of error theory. IEEE-754-2008. Classes single and double in Matlab. Machine accuracy. Errors of approximate values ​​of functions. Inverse error problem. Conditionality of the problem. Interpolation, Lagrange and Newton interpolation polynomials. Matlab function interp1. Numerical differentiation. Matlab function diff. Numerical methods for solving nonlinear equations and systems. Quadrature formulas of interpolation type. Matlab functions integral, trapz. Methods for estimating the residue. Generalization to multiple integrals. Construction of Gaussian formulas from the Jacobian matrix using the QR algorithm. Modifications of Gaussian formulas. Formulas of Radau and Lobato type. Kronrod schemes. Gauss-Turanian quadratures and generalizations. Convergence of quadrature processes. Trigonometric-type formulas. Integration of fast oscillatory functions. Interpolation cubic formulas. Review of cubic formulas for some special areas and certain weight functions. Numerical linear algebra. Gaussian elimination. LU factorization. Conditionality of systems of linear equations. Iterative methods. Functions linsolve, lu in Matlab. Approximation theory. Bernstein's theorem. Mean square approximation. Discrete mean square approximation. Chebyshev mini-max approximation. Implementation of linear and nonlinear regression in Matlab. ODJ. Cauchy problem. Euler's method. Convergence of methods. Crank-Nicholson method. Stability of methods. Stability on unbounded intervals. Higher order methods. Predictor corrector methods. ODE systems. Runge-Kut methods. ODE family implementation in Matlab. PDE. PDE classification. Elliptic equations. Variational formulation of the Dirichlet problem. Neumann problem. Finite difference method for elliptic equations. Finite element method for elliptic equations. Eigenvalue problem for elliptic equations. Parabolic equations. Variational formulation. Hyperbolic equations. Finite difference methods. Finite element methods. PDE toolbox in Matlab

The requirement for attending a course is defined by the curriculum of the study program.

1. M.M. Spalević, M.S. Pranić, Numerical Methods, Skver, Kragujevac, 2007. (http://mat.mas.bg.ac.rs) 2. G.V. Milovanović, M. Kovačević, M. Spalević, Numerical Mathematics - Collection of Solved Problems, University of Niš, 2003. (http://mat.mas.bg.ac.rs) 3. G.V. Milovanović, Numerical Analysis 1., 2., 3. parts, Scientific Book, Belgrade 1991. 4. B.S. Jovanović, Numerical Methods for Solving Partial Differential Equations, Modern Computing Technology and Its Applications, vol. 8, Math. Institute, Belgrade 1989., p. 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. Matlab software 11. Mathematica software 12. A.S. Cvetković, M.M. 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

(in Serbian) M. Spalević, M. Pranić, Numerical Methods, Skver, Kragujevac, 2007. ISBN 978-86-81829-84-4 (http://mat.mas.bg.ac.rs); (in Serbian) G.V. Milovanović, M. Kovačević, M. Spalević, Numerical Mathematics - Collection of Solved Problems, University of Niš, 2003. (http://mat.mas.bg.ac.rs) ISBN 86-80-135-70-4; (in Serbian) G.V. Milovanović, Numerical Analysis 1., 2., 3. parts, Scientific Book, Belgrade 1991. ISBN 86-23-20081-0; (in Serbian) B.S. Jovanović, Numerical methods for solving partial differential equations, Modern computing and its applications, vol. 8, Math. Institute, Belgrade 1989, p. 130. ISBN 86-80593-01-X; (in Serbian) A.S. Cvetković, M.M. Spalević, Numerical Methods, University of Belgrade, 2013. ISBN 987-86-7083-786-7