Course details

# Numerical Methods and Probability

INM Acad. year 2010/2011 Winter semester 5 credits

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.

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Assessment points

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Subject specific learning outcomes and competences

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.

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.

Prerequisites

- Mathematical Analysis (IMA)

Prerequisite kwnowledge and skills

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

Study literature

- Chapra, S.C., Canale, R.P.: Numerical Methods for Engineers. Fourth Edition. McGraw-Hill 2002, New York (the sample book can be borrowed from the teacher).
- Loftus, J., Loftus, E.: Essence of Statistics. Second Edition, Alfred A. Knopf, New York 1988 (the book can be borrowed from the technical library Brno, Kounicova Street).

Fundamental literature

- Ralston, A.: Základy numerické matematiky. Praha, Academia, 1978 (in Czech).
- Horová, I.: Numerické metody. Skriptum PřF MU Brno, 1999 (in Czech).
- Maroš, B., Marošová, M.: Základy numerické matematiky. Skriptum FSI VUT Brno, 1997 (in Czech).
- 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

Syllabus of lectures

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

Syllabus of numerical exercises

- Classical and geometric probabilities.
- Discrete and continuous random variables.
- Expected value and dispersion.
- Binomial distribution.
- Poisson and exponential distributions.
- Uniform and normal distributions, z-test.
- Mean value test, power.

Syllabus of computer exercises

- Nonlinear equation: bisection method, regula falsi, iteration, Newton method.
- System of nonlinear equtations, interpolation.
- Splines, least squares method.
- Numerical differentiation and integration.
- Ordinary differential equations, analytical solution.
- Ordinary differential equations, analytical solution.

Progress assessment

- Three 5-point homeworks: 15 points,
- five 3-point written tests: 15 points,
- final exam: 70 points.

Passing bounary for ECTS assessment: 50 points.

Controlled instruction

Three homeworks and five written tests.

Exam prerequisites

To pass homeworks and written tests with 5-point minimum.

Course inclusion in study plans