Numerical Methods and Probability

• 26 hrs lectures
• 13 hrs exercises
• 13 hrs pc labs

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

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 course yields fundamental knowledge from the probability theory (random event, probability, characteristics of random variables, probability distributions) which is necessary for simulation of random processes.

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

• 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

• 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.

• Principle of numerical methods, error classification, accuracy improvement.
• Metric space, completeness, contraction, Banach fixed- point theorem.
• Solving of nonlinear equations.
• Approximation, interpolation polynomial, least squares method, spline.
• Numerical derivative and integral, composite quadrature formulae.
• Solving of ordinary differential equations, one-step methods.
• Multi-step methods.
• Elementary event, operation with events, field of events.
• Definition of probability, conditional probability, event independence, total probability theorem.
• Random variable, distribution function, random variable distribution, probability density.
• Two-dimensional random variable, random variable characteristic.
• Some important distributions, law of large numbers, limit theorems.
• Fundamental concepts, hypothesis testing.

• Numerical error estimates, Richardson extrapolation.
• Interpolation polynomial.
• Application of Banach theorem.
• Probability.
• Distribution function, probability density.
• Normal distribution.
• Numerical characteristics.

• Solving of nonlinear equations.
• Approximation of functions.
• Spline.
• Numerical integration.
• Solving of differential equations.
• Fundamental types of probability distribution.

Either to pass the mid-term exam with the 10 points minimum, or the only one absence on practice classes.

Written mid-term exam, checked attendance on practice classes.