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Course, academic year 2018/2019
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Numerical Methods of Experimental Data Processing - NMAF035
Title in English: Numerické metody zpracování experimentálních dat
Guaranteed by: Institute of Physics of Charles University (32-FUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2004 to 2020
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: doc. RNDr. Jiří Bok, CSc.
RNDr. Ivan Barvík, Ph.D.
Classification: Physics > Mathematics for Physicists
Annotation -
Last update: T_FUUK (24.05.2004)
Basic and advanced numerical methods, used largely in the processing of experimental data
Aim of the course -
Last update: T_FUUK (18.05.2008)

see annotation

Teaching methods -
Last update: T_FUUK (18.05.2008)

lecture

Syllabus -
Last update: T_FUUK (24.05.2004)
1. NUMERICAL ERRORS, COMPUTATIONS ON COMPUTERS
History of the numerical mathematics. Absolute and relative error, significant digits. Truncation and rounding errors. The specialties of computer arithmetics. Cancellation, smearing, numerical instability, ill-conditioned problems.

2. SOLUTION OF NON-LINEAR EQUATIONS
Classification of equations. Direct and iterative methods. Method of bisection, iterations, Newton-Raphson and secant methods. Systems of non-linear equations.

3. NUMERICAL INTEGRATION
Newton-Cotes and Gauss methods. Richardson extrapolation and Romberg integration.

4. SYSTEMS OF LINEAR EQUATIONS
Problems of the linear algebra. Gauss elimination, LU and SVD decompositions. Condition number of matrix. Ill-conditioned problems.

5. LINEAR LEAST SQUARES
Approximation of functions (interpolation, Chebyshev approximation, least squares). "Derivation" of the least squares method from maximum likelihood principle. The system of normal equations.

6. WEIGHTED LINEAR LEAST SQUARES
Weights. Uncertainties of the computed parameters. The least squares in case of errors in both variables. Use of the SVD. Robust methods.

7. NON-LINEAR LEAST SQUARES
Linearization of certain special model functions and its pitfalls. Typical non-linear functions in optical spectroscopy. General minimization methods, Marquardt method. Application of random numbers for determination of parameters uncertainties.

8. RANDOM NUMBERS
Examples of random quantities, methods of random numbers generation. Tests of generators, chi-square test.

9. MONTE CARLO METHODS
Simulations, numerical problems.

10. FOURIER TRANSFORM
Fourier series, continuous and discrete Fourier transform. Gibbs phenomenon. Fourier transform of periodic and aperiodic functions. Sampling, Nyquist frequency, aliasing. Fast Fourier transform.

11. DECONVOLUTION
Influence of the measuring apparatus on the input signal (optical spectroscopy, astronomical photography), apparatus function. Methods of deconvolution: inverse Fourier transform, Van Cittert and Jansson methods. Maximum entropy method.

12. FACTOR ANALYSIS
History, classification.The principal component analysis and the "true" factor analysis. Mathematical methods, examples of applications.

 
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