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Numerical Methods

* Exchange students do not have to consider this information when selecting suitable courses for an exchange stay.

Course Unit Code470-2501/01
Number of ECTS Credits Allocated6 ECTS credits
Type of Course Unit *Compulsory
Level of Course Unit *First Cycle
Year of Study *Third Year
Semester when the Course Unit is deliveredWinter Semester
Mode of DeliveryFace-to-face
Language of InstructionCzech
Prerequisites and Co-Requisites
PrerequisitiesCourse Unit CodeCourse Unit Title
470-2210Numerical Linear Algebra 1
Name of Lecturer(s)Personal IDName
LUK76doc. Ing. Dalibor Lukáš, Ph.D.
Summary
In this course numerical methods for selected problems of mathematical analysis are tought. We shall also prove convergence rates and present efficient implementation.
Learning Outcomes of the Course Unit
Numerical methods stands behind computer solutions to complex engineering problems. The course Numerical Methods 1 aims at helping students to choose a proper algorithm for the solution of selected problems of Calculus and analyze the solution regarding stability (sensitivity of the output data on the inputs) and computational complexity.
Course Contents
Lectures:
Errors in numerical computations
Solution of non-linear equations: fixed point theorem, Newton method
Iterative solution of systems of linear equations
Eigenvalues and eigenvectors
Interpolation: polynomial, trigonometric, spline
Approximation:least square method, Tchebyshev metod
Numerical differentiation and quadrature
Numerical solution of initial value problem for ordinary differential equations

Projects:
The aim of the projects is solution of practical problem using numerical methods and their comparison with exact solution.
Project solution:
Problem analysis and proposal of appropriate numerical solution
Numerical solution
Exact solution and comparison with numerical solution
Discussion and conlusions

Excercises:
Introduction to Matlab
Error estimation on examples, computing of computer epsilon
Roots separation of nonlinear equations. Solution of nonlinear equations using bisection method, fixed point iterations and Newton method. Conditions of convergence. Solutions of systems of non-linear equations.
Jacobi and Gauss-Seidel nad SOR methods for solution of systems of linear equations.
Solution of systems of linear equations using steepest descent method and conjugate gradient method. Preconditioning.
Methods for finding of characteristic polynomial. Power method for largest and smallest eigenvalues.
Similarity transformations, Jacobi method, Givens, Housholder and Lanczos methods.
Lagrange and Newton interpolating polynomial, piecewise linear and cubic spline functions.
Least square method and normal equations. Systems of orthogonal functions.
Numerical differentiations.
Numerical quadrature: Newton-Cotes and Gauss formulae.
Numerical solution of initial value problem for ordinary differntial equations: Euler method, Runge-Kutta method.
Recommended or Required Reading
Required Reading:
- O. Steinbach, Numerische Mathematik 1. TU Graz, 2005.
- A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics. Springer, 2007.
- D. Lukáš, Numerické metody 1 - zápisky a programy z přednášek. VŠB-TU Ostrava, http://homel.vsb.cz/~luk76/Teaching/NM1/
- A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics. Springer, 2007.
Recommended Reading:
- W.H. Press, B.P. Flannery, S.A. Teukolski, W.T. Vetterling, Numerical Recipes in C. Cambridge University Press, 1990.
- V. Vondrák, L. Pospíšil, Numerické metody 1. VŠB-TU Ostrava, http://mi21.vsb.cz/modul/numericke-metody-1
- O. Steinbach, Numerische Mathematik 1. TU Graz, 2005.
Planned learning activities and teaching methods
Lectures, Tutorials, Project work
Assesment methods and criteria
Task TitleTask TypeMaximum Number of Points
(Act. for Subtasks)
Minimum Number of Points for Task Passing
Exercises evaluation and ExaminationCredit and Examination100 (100)51
        Exercises evaluationCredit40 (40)15
                Test 1Written test10 0
                Test 2Written test10 0
                ProjectProject20 0
        ExaminationExamination60 (60)11
                Practical examinationWritten examination40 0
                Theoretical examinationOral examination20 4