Title:

# Numerical Methods and Probability

Code:INM
Ac.Year:2014/2015
Sem:Winter
Curriculums:
ProgrammeField/
Specialization
YearDuty
IT-BC-3BIT2ndCompulsory
Language of Instruction:Czech
Credits:5
Completion:credit+exam (written)
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:26130130
ExamsTestsExercisesLaboratoriesOther
Points:7030000
Guarantor:Novák Michal, doc. RNDr., Ph.D. (DMAT)
Lecturer:Fuchs Petr, RNDr., Ph.D. (DMAT)
Novák Michal, doc. RNDr., Ph.D. (DMAT)
Instructor:Fuchs Petr, RNDr., Ph.D. (DMAT)
Novák Michal, doc. RNDr., Ph.D. (DMAT)
Svoboda Zdeněk, RNDr., CSc. (DMAT)
Faculty:Faculty of Electrical Engineering and Communication BUT
Department:Department of Mathematics FEEC BUT
Prerequisites:
 Discrete Mathematics (IDA), DMAT Mathematical Analysis (IMA), DMAT

Learning objectives:
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.
Description:
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.
Knowledge and skills required for the course:
Secondary school mathematics and some topics from Discrete Mathematics and Mathematical Analysis courses.
Learning outcomes and competencies:
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.
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.
Fundamental literature:

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

1. Fajmon, B., Hlavičková, I., Novák, M., Vítovec, J.: Numerická matematika a pravděpodobnost (Informační technologie), VUT v Brně, 2014 (elektronický učební text, plná verze) (in Czech)
2. Hlavičková, I., Hliněná, D.: Matematika 3 - sbírka úloh z pravděpodobnosti. Elektronický text FEKT VUT, VUT v Brně, 2010 (in Czech)
3. Novák, M.: Matematika 3 - sbírka příkladů z numerických metod. Elektronický text FEKT VUT, VUT v Brně, 2010 (in Czech)
Controlled instruction:
Ten written tests.
Progress assessment:

• 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 10-point minimum.