Programma del corso (english version)
Educational Goals | The course deals with modeling by numerical approximation, a technique important in many fields of data analysis. The use of this instrument enables us not only to process experimental data (included signals and images) but also it is the basis for the numerical integration and the simulation of the behavior of dynamical systems. These problems arise in many real world applications. |
Prerequisites | notions of numerical analysis concerning the numerical solution of linear and nonlinear systems, the LU and QR factorisations; notions of calculus concerning sequences, series and ordinary differential equations and systems; notions on eigenvalues and related features. |
Course syllabus | Interpolation: condition of Haar; polynomial interpolation (Lagrange form, divided differences and Newton form, Neville scheme, Taylor and Hermite polynomials); error of polynomial interpolation and choice of nodes; conditioning of the problem; example of Runge; theorems of Faber and Bernstein. Interpolation by spline functions. Bspline functions. Lemma of Schoenberg Whitney. Outline on the bivariate interpolation. Least squares approximation: normal equations and pseudoinverse of Moore-Penrose; numerical algorithms (the H’RK and SVD factorizations). Polynomial approximation (orthogonal polynomials). Continuous and discrete least squares approximation: Fourier series and trigonometric polynomials. Outline of Fourier trasform. Formulas of numerical derivation (Richardson extrapolation); stability. Formulas of numerical integration: degree of accuracy and stability; interpolatory formulas (Newton-Cotes), theorem of convergence (Polya); composite quadrature formulas and convergence. Romberg formulas and adaptive formulas; gaussian formulas. Outline on the multiple integration. Implementation and analysis of the methods in the framework of Scilab or Matlab. |
Reference books | L.W.Johnson, R.D. Riess: Numerical Analysis, second edition, Addison Wesley 1982; V.Comincioli - Analisi numerica - McGraw Hill, 1990; Burden R. L., Faires J.D., Numerical Analysis, Prindle Weber & Schmidt, Boston MA. 1985. |
Theaching activities | Ø conventional |
Exams | Ø written Ø oral |
