Course detail

Numerical Methods and Probability

FIT-INMAcad. year: 2018/2019

Numerical mathematics: Metric spaces, Banach theorem. Solution of nonlinear equations. Approximations of functions, interpolation, least squares method, splines. Numerical derivative and integral. Solution of ordinary differential equations, one-step and multi-step methods. Probability: Random event and operations with events, definition of probability, independent events, total probability. Random variable, characteristics of a random variable. Probability distributions used, law of large numbers, limit theorems. Rudiments of statistical thinking.

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

Students apply the gained knowledge in technical subjects when solving projects and writing the BSc Thesis. Numerical methods represent the fundamental element of investigation and practice in the present state of research.

Prerequisites

Secondary school mathematics and some topics from Discrete Mathematics and Mathematical Analysis courses.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

  • Ten 3-point written tests: 30 points,
  • final exam: 70 points.
    Passing bounary for ECTS assessment: 50 points.

    • Exam prerequisites:
    To pass written tests with at least 10 points.

    Course curriculum

    Not applicable.

    Work placements

    Not applicable.

    Aims

    In the first part the student will be acquainted with some numerical methods (approximation of functions, solution of nonlinear equations, approximate determination of a derivative and an integral, solution of differential equations) which are suitable for modelling various problems of practice. The other part of the subject yields fundamental knowledge from the probability theory (random event, probability, characteristics of random variables, probability distributions) which is necessary for simulation of random processes.

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

    Ten written tests.

    Recommended optional programme components

    Not applicable.

    Prerequisites and corequisites

    Not applicable.

    Basic literature

    Not applicable.

    Recommended reading

    Fajmon, B., Hlavičková, I., Novák, M., Vítovec, J.: Numerical Methods and Probability (Information technology), VUT v Brně, 2014
    Hlavičková, I., Hliněná, D.: Matematika 3. Sbírka úloh z pravděpodobnosti. VUT v Brně, 2015 (in Czech)
    Hlavičková, I., Novák, M.: Matematika 3 (zkrácená celoobrazovková verze učebního textu). VUT v Brně , 2014 (in Czech)
    Horová, I.: Numerické metody. Skriptum PřF MU Brno, 1999 (in Czech).
    Loftus, J., Loftus, E.: Essence of Statistics. Second Edition, Alfred A. Knopf, New York 1988.
    Maroš, B., Marošová, M.: Základy numerické matematiky. Skriptum FSI VUT Brno, 1997 (in Czech).
    Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability for Engineers. Third Edition. John Wiley & Sons, Inc., New York 2003
    Novák, M.: Matematika 3 (komentovaná zkoušková zadání pro kombinovanou formu studia). VUT v Brně, 2014 (in Czech)
    Novák, M.: Mathematics 3 (Numerical methods: Exercise Book), 2014
    Ralston, A.: Základy numerické matematiky. Praha, Academia, 1978 (in Czech).
    Taha, H.A.: Operations Research. An Introduction. Fourth Edition, Macmillan Publishing Company, New York 1989.

    Classification of course in study plans

    • Programme IT-BC-3 Bachelor's

      branch BIT , 2 year of study, winter semester, compulsory

    Type of course unit

     

    Lecture

    26 hod., optionally

    Teacher / Lecturer

    doc. RNDr. Michal Novák, Ph.D.
    RNDr. Petr Fuchs, Ph.D.

    Syllabus

    1. Introduction to numerical methods.
    2. Numerical solution of linear systems.
    3. Numerical solution of non-linear equations and systems.
    4. Approximation, interpolation.
    5. Numercial integration and differentiation.
    6. ODE's: Introduction, numerical solution of first-order initial value problems.
    7. Introduction to statistics, vizualization of statistical data.
    8. Introduction to probability theory, probability models, conditional and complete probability.
    9. Discrete and continuous random variables.
    10. Selected discrete distributions of probability.
    11. Selected continuous distributions of probability.
    12. Statistical testing.
    13. Reserve, revision, consultations.

    Computer-assisted exercise

    26 hod., compulsory

    Teacher / Lecturer

    doc. RNDr. Zdeněk Svoboda, CSc.
    RNDr. Petr Fuchs, Ph.D.
    doc. RNDr. Michal Novák, Ph.D.

    Syllabus

    1. Nonlinear equation: Bisection method, regula falsi, iteration, Newton method.
    2. System of nonlinear equtations, interpolation.
    3. Splines, least squares method.
    4. Numerical differentiation and integration.
    5. Ordinary differential equations, analytical solution.
    6. Ordinary differential equations, analytical solution.