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# PROG1

**Course:** Computer programming and numer. methods

**Department/Abbreviation:** SLO/PROG1

**Year:** 2021

**Guarantee:** 'Ing. Jaromír Křepelka, CSc.'

**Annotation:** The aim consists in application of mathematical analysis and algebra knowledge for understanding the basic numerical methods usable in scientific calculations and practically demonstrate how these algorithms work when implemented in the computer environment.

**Course review:**

- Computation errors - influence of finite number of digits on the accuracy of the computation
- 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, zero points of the polynomial expressions (Lin- Bairstow method, method of Siljakov coefficients, Laguerre method), eigenvalues and eigenvectors of the matrices (general problem, symmetric matrices, LU and QR algorithms)
- 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
- Interpolation, numerical differentiation and integration - Laguerre polynomials, the best trigonometric polynomials, Fourier series, discrete and fast Fourier transform, cubic splines, Čebyšev approximation (Remez algorithm), numerical differentiation and integration (trapezoidal formula, Newton-Cotes quadrature formula, Simpson formula, Gauss methods, special formula)
- Numerical solutions of ordinary differential equations - problem with initial condition (Euler method, Runge-Kutta methods, Merson method, automatic choice of integration step, implicit integration methods, stability convergence, correctness), boundary problem (method of shooting, linear systems of differential equations, analytical solutions, problems of existence of numerical solutions, construction of difference schemes, Marcuk identity)
- 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)
- Basics of numerical solving of partial differential equations - difference scheme, fully conservative difference scheme