Course detail

Mathematics 3

FEKT-KMA3Acad. year: 2011/2012

Numerical mathematics: numerical errors, functional approximation, interpolation and splines, least squares method, numerical derivation and integration, numerical solving of differential equations.
Probability: events and probabilities, conditional probability, independent events, the Bayes theorem, random variable, probability distributions (chosen types), standard normal distribution.
Mathematical statistics: statistical measures, tests.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Michal Novák, Ph.D.

Department

Department of Mathematics (UMAT)

Learning outcomes of the course unit

After the completion of the course the students should be able to solve equations and systems of equations numerically, use the least squares method and interpolation, numerical derivation and integration formulas as well as solve some types of differential equations numerically.In the area of probability they should know where to use specific probabilistic models. Some statistical test are also considered.

Prerequisites

Knowledge of combinatorics at secondary school level; KMA1 and KMA2 courses.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Written examination is evaluated by maximum 63 points, student's work during the semestr is assesed by maximum 37 points.

Course curriculum

Five tutorials and two computer classes:

Tutorial 1: Introduction, Numerical methods I
Tutorial 2: Numerical methods II
Computer class 1: Introduction to Matlab, examples from numerical methods
Tutorial 3: Numerical methods completed, probability models
Tutorial 4: Some distributions of probability, introduction to statistical testing
Computer class 2: Examples from probability and statistics
Tutorial 5: Repetition, seminar, information about the exam.

Work placements

Not applicable.

Aims

Subject comprises of two mathematical disciplines: NUMERICAL METHODS whose objective is to introduce fundamentals of numerical problem solving, and PROBABILITY whose purpose is to consider probabilistic techniques in problem solving.

Specification of controlled education, way of implementation and compensation for absences

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

B. Fajmon, I. Hlavičková, M. Novák, J. Vítovec, Numerická matematika a pravděpodobnost (Informační technologie), VUT v Brně, 2014 (elektronický učební text, plná verze) (CS)
HLAVIČKOVÁ, I.; HLINĚNÁ, D. Matematika 3 - Sbírka úloh z pravděpodobnosti. Matematika 3 - Sbírka úloh z pravděpodobnosti. Brno: UMAT FEKT VUT, 2007. s. 1-77. (CS)
NOVÁK, M. Matematika 3: Sbírka úloh z numerických metod. Brno: FEKT VUT, 2010. (CS)

Recommended reading

Haluzíková, A.: Numerické metody. Skriptum FEI VUT Brno, 1989. (CS)
Zapletal, J.: Základy počtu pravděpodobnosti a matematické statistiky. Skriptum FEI VUT Brno, PC-DIR 1995. (CS)

Classification of course in study plans

  • Programme EEKR-BK Bachelor's

    branch BK-AMT , 2. year of study, winter semester, compulsory
    branch BK-EST , 2. year of study, winter semester, compulsory
    branch BK-MET , 2. year of study, winter semester, compulsory
    branch BK-SEE , 2. year of study, winter semester, compulsory
    branch BK-TLI , 2. year of study, winter semester, compulsory

  • Programme EEKR-CZV lifelong learning

    branch ET-CZV , 1. year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

1. Banach theorem. Jacobi and Gauss-Seidel iterative methods.
2. Interpolation, least squares method.
3. Spline, numerical methods of differentiation.
4. Numerical integration - trapezium and Simpson methods.
5. Solving ODE - Euler method and modifications of the method. Runge - Kutta method.
6. Solving ODE - Euler method for a system of equations, shooting method, finite difference method. Multistep methods.
7. Probabilistic models (classical and geometrical probabilities, discrete and continuous random variables).
8. Expected value and dispersion.
9. Binomial distribution. Fundamentals of statistical tests. The sign test.
10.Poisson and exponential distributions. Their application in queueing theory.
11.Normal distribution. Central limit theorem. Approximation of binomial distribution by means of normal distribution. Z-test and power.
12.The mean expected value test.

Exercise in computer lab

14 hours, compulsory

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

There are no computer classes in KMA3. For reference purposes, BMA3 classes are given:

1. Root separation, bisection, regula falsi.
2. Iterative metod, Newton method.
3. Systems of nonlinear equations, interpolation.
4. Spline, least squares method.
5. Numerical differentiation and integration.
6. Numerical methods for ordinary differential equations - Euler method, Runge - Kutta method, finite difference method.

The other activities

12 hours, compulsory

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

Classes have the form of tutorials; BMA3 practical classes are as follows:

1. Classical and geometrical probability.
2. Discrete and continuous random variable.
3. Expected value and dispersion.
4. Binomial distribution. The sign test.
5. The Poisson and exponential distributions, queuing theory.
6. Uniform and normal distributions, binomial approximation of normal distribution, z-test.
(7. Mean expected value test, power.)