## Numerical Methods

### Studijní plán: Aplikovaná technika pro průmyslovou praxi - navazující kombinovaná forma, platný pro studenty, kteří započali studium od ZS 2021/2022

Předmět Numerical Methods (NM-1) Katedra matematiky (KM) doc. RNDr. Petr Gurka, CSc. Anglicky 3
Prezenční studium
Přednáška1 h
Cvičení2 h
Kombinované studium
Tutoriál / přednáška6 h
Cvičení10 h
Studijní plán Typ Sem. Kred. Ukon.
Aplikovaná technika pro průmyslovou praxi - navazující kombinovaná forma, platný pro studenty, kteří započali studium od ZS 2021/2022 P 1 3 kr. KZ
Aplikovaná technika pro průmyslovou praxi - navazující platný pro studenty, kteří započali studium od ZS 2021/2022 P 1 3 kr. KZ

#### Anotace

The course acquaints students with the main types of numerical calculations that are used in engineering practice and are the basis of computational software tools. It is mainly the approximation of values ​​of functions or irrational numbers using only basic arithmetic operations, approximate solution of nonlinear equations and their systems, finding extrema, solving large systems of linear equations, approximation and interpolation of data, approximate calculation of a definite integral and numerical solution of differential equations. Students will get acquainted with the theoretical properties of the relevant methods and at the same time learn to use these methods independently to solve simple (practical) problems. Students can use the Matlab program or another programming environment to perform calculations. The advantage is previous completion of mathematical subjects and mechanics, but it is not a condition.

Knowledge: The student will gain an overview of basic numerical procedures in several areas of numerical calculations: approximation problems, solution of nonlinear equations, solution of large systems of linear equations, numerical derivation and integral and numerical solution of differential equations. Part of the acquired knowledge is awareness of the computational complexity and accuracy of individual methods and the pitfalls of implementing appropriate algorithms.

Skills: The student is able to numerically solve a simple problem from the discussed areas, so he can choose a suitable method, ev. edit it and write the code separately in the selected programming environment. The student is able to compare the complexities of various calculations and the accuracy of the obtained results. Especially the student can numerically solve nonlinear equations of one variable and optimization problems of one or more variables, use stationary and iterative methods to solve systems of linear equations, approximate data using the least squares method, calculate the approximate integral and obtain an approximate solution of an ordinary differential equation with initial condition or with boundary conditions. The student becomes able to understand the software components that use these algorithms, and create these components for basic numerical algorithms.

Course syllabus:

• Calculations with finite errors, approximation of real numbers, Taylor expansion.

• Methods of numerical solution of nonlinear equations, bisection method, Newton method, method of false position, tangent method, secant method.

• Numerical search for extremes.

• Solutions of linear systems – direct methods: Gauss elimination, Gauss-Jordan elimination.

• Norm of a vector and of a matrix, conditionality of a matrix.

• Solutions of linear systems – iterative methods: Jacobi method, Gauss-Seidel method, maximum slope method.

• Polynomial interpolation of given data.

• Method of least squares, polynomial or trigonometric approximation. Introduction to Signal Processing.

• Numerical integration: rectangular, trapezoidal and Simpson rule. Gaussian quadrature. Numerical calculation of the centre of gravity of a planar shape.

• Numerical differentiation: approximation of the derivative of a function.

• Numerical solution of ordinary differential equations. Initial value problem for 1st and 2nd order equations, Euler method. Equations of motion.

• Numerical solution of boundary value problem for 2nd order equation, finite-difference method. Beam deflection.

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