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Budapesti Műszaki és Gazdaságtudományi Egyetem
Közlekedésmérnöki és Járműmérnöki Kar
1. Tantárgy neve Numerical optimization
2. Tantárgy angol neve Numerikus optimalizálás
3. Tantárgykód BMEKOVRM334 4. Követelmény vizsga 5. Kredit 5
6. Óraszám 3 (16) Előadás 0 (0) Gyakorlat 1 (5) Labor
7. Tanterv
Logisztikai mérnöki mesterképzési szak (L)
 8. Szerep
Kötelező (k) a Logisztikai mérnöki mesterképzési szakon (L)
9. A tantágy elvégzéséhez szükgésges tanulmányi munkaóra összesen 150
Kontakt óra 56 Órára készülés 13 Házi feladat 28
Írásos tananyag 38 Zárthelyire készülés 0 Vizsgafelkészülés 15
10. Felelős tanszék Vasúti Járművek, Repülőgépek és Hajók Tanszék
11. Felelős oktató Dr. Rohács József
12. Oktatók Dr. Bicsák György
13. Előtanulmány  
14. Előadás tematikája
Introduction: scope of lectures, content and requirements. System analysis, model generation, modelling and simulation. General models, simplifications. Source of errors, model types and solution possibilities. Analytic, geometric and numerical solutions.
Functions, vectors, matrices, basic operations. Classical and floating-point error-calculation. Sensitivity and numerical stability. Investigation of solution technics. Representing the solutions, evaluation.
Solution of system of equations. Single variable, non-linear equations. Successive approximation, Newton iteration and secant method. Solution of polynomial equation. Horner method and Newton-method.
Numerical solution of linear system of equations. Gauss-elimination and LU decomposition. Numerical solution of Eigenvalue problem.
Extremum problems, optimization. Linear programming, transforming to standard form. Simplex method, dual simplex method. Optimization of non-linear functions. Non-linear programming. Sensitivity analysis, multipurpose linear programming. Goal and object dependent optimisation. Optimisation by using soft-computing techniques. Gradient method. Examining specific cases, optimization tasks in logistics systems and processes. Fundamentals of game theory.
Functions, series of functions, approximation. Taylor series, MacLaurin series, Fourier series.
Polynomial-interpolation, Newton, Lagrange and Hermite interpolation. Application of Splines. Generating curves and surfaces with using Splines. Bezier polynomials, NURBS surfaces. Approximation, Chebyshev and Padé approximation. Harmonical analysis, fast Fourier transformation (FFT).
Numerical differentiation, integration. Approximation of derivatives using finite difference method. Approximation of derivatives using Lagrange and Newton interpolation formulas. Numerical integration, general quadrature formula. Trapezoidal and Simpson formula. Romberg iteration.
Initial value problems, ordinary differential equations. Explicit formulas: Euler method, 4th order Runge-Kutta method. Implicit formulas, predictor-corrector methods.
Approximation of partial differential equations. Boundary conditions, finite difference method, finite volume method, finite element method.
Stochastic process modelling. System input data generation. Monte-Carlo simulation.
15. Gyakorlat tematikája
 
16. Labor tematikája
MATLAB application of the introduced methods.
17. Tanulási eredmények
A. Tudás
  • knowing the fundamentals of numerical approximation methods used in engineering instead of analytic algorithms. Knowing to find and apply the most suitable numerical method for a certain problem.
B. Képesség
  • can implement different algorithms to a programming language and to find the best approximation method for a given mathematical problem.
C. Attitűd
  • interested, responsive.
D. Önállóság és felelősség
  • can work individually and in teamwork.
18. Az aláírás megszerzésének feltétele, az aláírás érvényessége
2 midterm exams from the theoretical part, 50 points / exam.
1 project work for a group of 4-5 students, for n*100 points (n is the number of students). The points can be divided between the group members according to their wish.
Grade calculation: summing all the points, the total points gives the final grade as follows: 0 – 79 - 1; 80 – 109 - 2; 110 – 139 - 3; 140 – 169 - 4; 170 – 5
19. Pótlási lehetőségek
Because of the point-collection system, no minimum points are determined for the midterm exams or for the project work. The retake possibilities are the following: on the replacement week the 1st midterm exam, or the 2nd midterm exam can be tried again for 50 points, or a combined 1st+2nd midterm exam retake for 100 points.
20. Jegyzet, tankönyv, felhasználható irodalom
Examples, documents and training materials, given out during lectures, presentations.
György Bicsák, Dávid Sziroczák, Aaron Latty: Numerical Methods
Ramin S. Esfandiari: Numerical methods for engineers and scientists using MATLAB, ISBN 978-1-4665-8570-6
Erwin Kreyszig: Advanced engineering mathematics, 10th edition, ISBN 978-0-470-45836-5
Tantárgyleírás érvényessége 2019. október 10. Jelen TAD az alábbi félévre érvényes Inactive courses