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Course: Numerical methods for physicists 1

Department/Abbreviation: SLO/NMF1

Year: 2020

Guarantee: 'Ing. Jaromír Křepelka, CSc.'

Annotation: The aim consists in application of mathematical analysis and algebra knowledge to psysical examples and demonstrate how these algorithms work when they are implemented in the computer environment.

Course review:
1. Computation errors - influence of finite number of digits on the accuracy of the computation. 2. Algebraic methods - systems of linear algebraic equations (systems with non-empty null-space, predetermined systems), three-diagonal scheme, Gauss and Gauss-Jordan method, LU decomposition, inversion of matrices . 3. Eigenvalues and Eigenvectors of Matrices - general problem, symmetric matrices, LU and QR algorithms, iteration algorithms. 4. Roots of Polynomials - Lin- Bairstow method, method of Siljakov coefficients, Laguerre method. 5. Solving of systems of nonlinear equations - bisection of the interval, Newton method of the tangents, Richmond method of tangential hyperboles, their generalization for the systems of equations, Čebyšev iteration methods, Warner scheme (generalized method of tangents), gradient methods. 6. Interpolation - Laguerre polynomials, the best trigonometric polynomials, cubic splines, Čebyšev approximation (Remez algorithm), Fourier series. 7. Numerical differentiation and integration - trapezoidal formula, Newton-Cotes quadrature formula, Simpson formula, Gauss methods, special formula. 8. Minimization of functions and optimization - minimization of functions of one variable (golden section, differential methods), simplex method of minimization of functions of more variables, gradient methods (method of conjugated vectors, Powell quadratic convergent method), linear programming, combinatory problems (permutation problems - lexicographical selection, problem of traveling salesman, method of simulative annealing, evolution algorithms - self-organized migration algorithms).