Interpolation and approximation methods. Hermite interpolation also for curves. Splines functions. Bernstein approximation. Least square approximation. Numerical derivation. Quadrature rules in general and interpolatory quadrature rules. Matlab.
L. Gori, Calcolo Numerico IV edizione 2006, Edizioni KAPPA
F. Fontanella, A. Pasquali, Calcolo numerico, Metodi e algoritmi. Vol. 2, 1980, Pitagora Editore
Learning Objectives
To be able to identify and solve an approximation problem.
Prerequisites
Foundamentals of numerical analysis. The language MATLAB
Teaching Methods
Lectures by the teacher and exercises in the lab.
Type of Assessment
Oral exames and Matlab program
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;