Syllabus
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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.
Annotation
Basic and advanced numerical methods, used largely in the processing of experimental data