COURSE CONTENT:
Approximate numbers: sources of errors, significant figures, rounding numbers, errors of arithmetic operations and functions, error progression. Nonlinear equations: root isolation, bisection method, Newton-Raphson method, secant method, method of successive approximations. Interpolation: interpolation problem, finite differences, Newton's method, Lagrange's method, spline method. Numerical differentiation and integration: numerical differentiation of continuous and discrete functions, numerical integration, trapezoidal formula, Simpson's formula. Ordinary differential equations: Euler's method, Runge-Kutta methods, finite difference method. Optimization: non-derivative and derivative minimisation methods, simplex, steepest descents algorithm, conjugate gradients algorithm, Newton-Raphson method, global search, Monte Carlo method, genetic algorithm. Probability theory: classical definition of probability, axiomatic definition of probability, conditional probability, total probability, Bayes formula, basics of combinatorics, fundamental theorem of counting, variations, permutations, combinations. Basic statistics: descriptive statistics, measures of central tendency and dispersion, sampling and graphical representation of data. Discrete randrom variables: random variables, probability function, cumulative distribution function, moments of distribution, uniform distribution, Bernoulli trials, binomial distribution, Poisson distribution, hypergeometric distribution, estimate of distribution parameters. Continuous distribution function: probabiltiy density function, cumulative distribution function, moments of distribution, continuous uniform distribution, Gauss distribution, exponential distribution, estimate of distribution parameters. Statistical hypothesis testing: null-hypothesis, statistical model checking, location and dispersion tests. Regression: linear regression and correlation, confidence intervals, nonlinear regression.
LEARNING OUTCOMES:
- to discriminate the exact and the approximate numbers
- to calculate the relative and the absolute error
- to solve nonlinear equations using adequate numerical methods
- to use numerical methods for interpolation
- to use numerical methods for differentiation and integration
- to discriminate numerical methods for optimisation of functions
- to explain basic principles of probability theory
- to explain basic principles of statistics
- to discriminate discrete and continuous variables
- to explain probability density function and cumulative distribution function
- to use binomial, Poisson, and hypergeometric distributions
- to use normal (Gaussian) distribution and uniform distribution
- to define statistical tests and hypothesis
- to use regression analysis
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