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.
Course notes written by the students.
Learning Objectives
To be able to identify and solve a numerical problem with particular attention to an approximation problem. To be able to identify algorthims for the solution.
Prerequisites
Foundamentals of numerical analysis. The language MATLAB
Teaching Methods
Lectures by the teacher and Matlab exercises in the lab.
Type of Assessment
Oral discussion to verify a critical attitude and a transversal hability to solve problems also by implementing algorithms in Matlab
Course program
[1] Approximation and Interpolation:
[1.1] Position of the problem: possible spaces of functions and types of approximants;
[1.2] Lagrange polynomial interpolation; analysis of the error;
The case of uniform knots and the corresponding asymptotic behaviour;
[1.4] Stability in interpolation. The Lebesgue constant;
[1.5] Chebyshev's polynomials; interpolation with knots given by the zeros of Chebyshev polynomial;
[1.6] Weierstrass' theorem and Bernstein polynomials;
[1.7] Hermite interpolation; Error analysis;
[1.8] Splines functions: definition, properties and the base of truncated powers;
[1.9] Splines interpolating and approximating; cubic splines interpolating at the knots;
[1.10] B-splines and De Boor algorithm (in particular for the cubic case);
[1.11] The parametric case: interpolation with uniform parametrization and with arc-length parametrization;
[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;