Course detail

Numerical Methods and Probability

FIT-INMAcad. year: 2014/2015

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

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 Bc. 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

To pass written tests with 10-point minimum.

Course curriculum

    Syllabus of lectures:
    1. Banach theorem. Iterative methods for linear systems of equations.
    2. Interpolation, splines.
    3. Least squares method, numerical differentiation.
    4. Numerical integration: trapezoid and Simpson rules.
    5. Ordinary differential equations, analytical solution.
    6. Ordinary differential equations, numerical solution.
    7. Test 1 (15 points).
    8. Probability models: classical and geometric probabilities, discrere and continuous random variables.
    9. Expected value and dispersion.
    10. Poisson and exponential distributions.
    11. Uniform and normal distributions. Central limit theorem, z-test, power.
    12. Mean value test.
    13. Test 2 (15), review.

    Syllabus of numerical exercises:
    1. Classical and geometric probabilities.
    2. Discrete and continuous random variables.
    3. Expected value and dispersion.
    4. Binomial distribution.
    5. Poisson and exponential distributions.
    6. Uniform and normal distributions, z-test.
    7. Mean value test, power.

    Syllabus of computer exercises:
    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.

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

Ralston, A.: Základy numerické matematiky. Praha, Academia, 1978. Horová, I.: Numerické metody. Skriptum PřF MU Brno, 1999. Maroš, B., Marošová, M.: Základy numerické matematiky. Skriptum FSI VUT Brno, 1997. Loftus, J., Loftus, E.: Essence of Statistics. Second Edition, Alfred A. Knopf, New York 1988. Taha, H.A.: Operations Research. An Introduction. Fourth Edition, Macmillan Publishing Company, New York 1989. Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability for Engineers. Third Edition. John Wiley & Sons, Inc., New York 2003.

Recommended reading

Fajmon, B., Hlavičková, I., Novák, M., Vítovec, J.: Numerická matematika a pravděpodobnost (Informační technologie), VUT v Brně, 2014 Hlavičková, I., Hliněná, D.: Matematika 3. Sbírka úloh z pravděpodobnosti. VUT v Brně, 2015 Hlavičková, I., Novák, M.: Matematika 3 (zkrácená celoobrazovková verze učebního textu). VUT v Brně , 2014 Novák, M.: Matematika 3 (komentovaná zkoušková zadání pro kombinovanou formu studia). VUT v Brně, 2014 Novák, M.: Mathematics 3 (Numerical methods: Exercise Book), 2014

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.

Syllabus

  1. Banach theorem. Iterative methods for linear systems of equations.
  2. Interpolation, splines.
  3. Least squares method, numerical differentiation.
  4. Numerical integration: trapezoid and Simpson rules.
  5. Ordinary differential equations, analytical solution.
  6. Ordinary differential equations, numerical solution.
  7. Test 1 (15 points).
  8. Probability models: classical and geometric probabilities, discrere and continuous random variables.
  9. Expected value and dispersion.
  10. Poisson and exponential distributions.
  11. Uniform and normal distributions. Central limit theorem, z-test, power.
  12. Mean value test.
  13. Test 2 (15), review.

Fundamentals seminar

13 hod., optionally

Teacher / Lecturer

doc. RNDr. Michal Novák, Ph.D.

Syllabus

  1. Classical and geometric probabilities.
  2. Discrete and continuous random variables.
  3. Expected value and dispersion.
  4. Binomial distribution.
  5. Poisson and exponential distributions.
  6. Uniform and normal distributions, z-test.
  7. Mean value test, power.

Exercise in computer lab

13 hod., optionally

Teacher / Lecturer

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.