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Lecturer(s)
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Černá Dana, doc. RNDr. Ph.D.
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Hozman Jiří, RNDr. Mgr. Ph.D.
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Course content
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Lectures: 1. Numerical methods - numerical model, sources of error, numerical stability, computation speed. 2. Direct methods for solving linear systems - Gaussian elimination, Gaussian elimination for tridiagonal matrix, LU decomposition, Cholesky decomposition. 3. Iterative methods for solving linear systems - Jacobi, Gauss-Seidel. 4. Interpolation - Lagrange and Hermite interpolation, existence and uniqueness, error, Runge phenomenon, Lagrange and Newton form. 5. Spline interpolation, existence and uniqueness, linear and cubic spline. Trigonometric interpolation, Fourier transform. 6. Function approximation using the least square method, a system of normal equations. 7. Numerical integration - rectangular rule, trapezoidal rule, and Simpson's rule. 8. Nonlinear equations - bisection method, Newton method, secant method. 9. Numerical solution of ordinary differential equations with initial conditions - existence and uniqueness, a transformation of n-th order differential equation into a system, error types, and method order. Euler methods, the interval of absolute stability. 10. Runge-Kutta methods. Implicit Euler methods. 11. Multistep methods - Adams-Bashforth and Adams-Moulton methods. 12. Second-order ordinary differential equations with boundary conditions - finite difference method. 13. Classification of second-order partial differential equations. 14. Numerical solution of elliptic partial differential equations - finite difference method.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing), Written assignment presentation and defence
- Class attendance
- 56 hours per semester
- Preparation for exam
- 45 hours per semester
- Semestral paper
- 20 hours per semester
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Learning outcomes
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The subject focuses on the theory and implementation of basic numerical methods, e.g., direct and iterative methods for solving linear equations, polynomial interpolation, numerical integrations, numerical methods for ordinary and partial differential equations.
Knowlige of fundamentals of numerical mathematics.
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Prerequisites
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Passing of mathematical lectures.
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Assessment methods and criteria
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Oral exam, Written exam
Credit: Working out a semestral work. Exam: Written.
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Recommended literature
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Brzezina M., Dvořák M., Kalousek Z., Salač P., Staněk J. Šimůnková M.:. Matematika IV. Liberec, 1996. ISBN 80-7083-19-2.
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SMITH, D. M.:. Engineering computation with MATLAB.. Boston: Pearson/Addison Wesley, 2007. ISBN 0-321-48108-9.
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Ueberhuber, Ch. W.:. Numerical Computation 1, 2.. Berlin, Springer-Verlag, 1997.
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Vitásek, E.:. Numerické metody.. Praha, SNTL, 1987.
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Vitásek, E.:. Vybrané kapitoly z teorie numerických metod pro řešení diferenciálních rovnic.. Plzeň: ZČU, 2002. ISBN 80-7082-857-9.
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