Academic Year 2021-2022

APPROXIMATION THEORY AND PRACTICE

Teachers

Rossana Vermiglio
Unit Credits
6
Teaching Period
First Period
Course Type
Supplementary
Prerequisites. First two years of Laurea Triennale in Mathematics
Teaching Methods. Lessons and laboratory exercises with Matlab and/or Chebfun https://www.chebfun.org. ‘English-taught course’ when there are international students.
Verification of Learning. Oral examination on the fundamental theorems with proofs and on the computational properties of the algorithms presented in the course
More Information. The course will be face-to-face but the students will have available the recordered lectures of the past academic years.
Objectives
The course provides the students with knowledge on the fundamental results and properties of the basic numerical methods to approximate functions, integrals and derivatives. The course includes a brief introduction to MATLAB, a mathematical software widely used in many research fields and in the work environment, and some laboratory activities to solve simple approximation problems, to experimentally analyze the properties of the numerical results and to compare the performance of different algorithms. Such skills aim to develop the maturity of judgment and the critical sense.

The acquired skills allow to continue studying numerical analysis at an advanced level, and provide the students with mathematical tools useful in various application fields. In fact the problems treated in the course arise, for instance in computer science, natural and social sciences, engineering, medicine, biology, and economics and financial mathematics.

Contents
– Approximation of data and functions: introduction to interpolation, polynomial interpolation, piecewise polynomial interpolation, splines interpolation, trigonometric interpolation, FFT, parametric interpolation; Bezier curves, B-splines curves. Best uniform approximation (Chebyshev theory); best approximation in prehilbertian spaces (Fourier theory). Orthogonal polynomials.

– Numerical integration: Newton-Cotes formulas, Gaussian formulas, composite and adaptive formulas, Richardson extrapolation and Romberg integration, singular integrals.

– Numerical derivation: finite differences, pseudo-spectral techniques. -The theoretical results are complemented with MATLAB and/or Chebfun laboratory activities on simple case studies.

Texts
– Notes of the teacher available on e-learning

-A. QUARTERONI. R. SACCO , F. SALERI I: Numerical Mathematics, 2 ed. Springer Verlag (2007)

-R. Bevilacqua, D. Bini, M. Capovani, O. Menchi, Metodi Numerici, Zanichelli (1992).

-L.N. Trefethen Approximation Theory and Approximation Practice, Extended Edition SIAM Ed (2020)