Approssimation and interpolation: fro classical methods to B-splines.
rectangular liner systems: the linear problem of least square.
Numerical derivation: soma basic and simple ideas.
Quadrature rules..
MATLAB: its use for the solution of more complex problems connected with approximation and interpolation.
[1] D. Bini, M. Capovani, O. Menchi, Metodi numerici per l'algebra lineare, Zanichelli 1988
[2] L. Gori, Calcolo Numerico, Edizioni Kappa, 1999,
[3] M.L. Lo Cascio, Fondamenti di Analisi Numerica, McGraw-Hill, 2008
[4] M.G.Gasparo, R. Morandi, Elementi di Calcolo Numerico:metodi e algoritmi,
McGraw-Hill editore, 2008
Learning Objectives
The idea is to investigate more advanced but classical methods of numerical analysis also by the help of a computer. The students are request to consider more sofisticated algorithms to solve complex problems originating from different type of applications.
Prerequisites
Basic knowledge of Numerical Analysis
Teaching Methods
Lectures and pratical treaning in the Lab.
Type of Assessment
Oral exams and a Matlab implementation for the solution of a tprescibed problem.
Course program
[1] Approximation and Interpolation:
[1.1] Position of the problem and possible solutions;
[1.2] Polynomial interpolation via Lagrange form; Analysis of the error;
[1.3] The error in the case of uniform knots and discussion about the asymptotic behaviour;
[1.4] Stability of the interpolation: the Lebesgue constant;
[1.7] Osculating polynomials and Hermite interpolation; The error in Hermite interpolation;
[1.8] Splines: definition, basic properties and the base of truncated power;
[1.9] Splines interpolating and approximating; cubic splines interpolating at the knots (natural and complete)
[1.10] B-splines: the perfect base for splines: recurrence formula of C. De Boor (particular attention to the cubic case);
[1.11] The paramteric case: parametric interpolation and the problem of parameters selection: the uniform and the arc-length parametrizations;
[2] Rectangular linear systems: the solution of an ordinary least squares problem
$$\min_{x\in \RR^n}\|Ax-b\|_2,\ A\in \RR^{m\times n},\ b\in \RR^n,\ m\ge n$$
[2.1] Existence and unicity of the solution;
[2.2] Solution via normal equations $A^TAx=A^Tb$;
[2.3] Orthogonal matrices: the Householder matrices ;
[2.4] $QR$ factorization;
[2.5] Solution of the least squares problem with $QR$;
[2.7] Best trigonometric approximation and the special case of trigonometric interpolation; Fourier;
[3] Numerical derivation: some simple and basic ideas. The method of un-dertermined coefficients;
[4] Quadrature rules (FdQ)
[4.1] Position of the problem. The linear case with knots $x_0,\cdots,x_n$ of type $\sum_{i=0}^n\omega_if(x_i)$
[4.2] Degree of precision $\nu$ for a FdQ (GdP); limitation from above $\nu\le 2n+1$
[4.3] Convergence of FdQ to the integral $n\rightarrow \infty$. Analysis of the stability;
[4.4] The method of un-dertermined coefficients
[4.5] Interpolatory FdQ : analysis of the GdP (limitation from above and from below $n\le \nu\le 2n+1$);
[4.5.1] Closed Newton-Cotes; GdP and examples;
[4.5.2] Open Newton-Cotes; GdP and examples;
[4.5.3] Generalized Newton-Cotes formulas; Trapezoidal and Simpson;
[4.5.4] Practical evaluation of the error with the Richardson extrapolation;
[4.5.5] Adaptive FdQ;