B018972 - ADVANCED NUMERICAL ANALYSIS

Italian

Knowledge and understanding: acquisition of the theoretical and algorithmic bases of numerical analysis with particular attention to numerical linear algebra and to its basic problems. Starting to understand the nature of numerical methods and the modalities of research in this field.
- Practical application of acquired knowledge: ability to write numerical algorithms for solving mathematical problems and implement them in Matlab language.

Numerical Methods for eigenvectors and eigenvalues. Singular value decomposition. Numerical mathods for initial value problems. Matlab

Basic of numerical analysis, Linear algebra and Calculus

Lectures and laboratory training. The participation to laboratory training is compulsory

Oral exam made of an open discussion aimed at evaluating also the capability of linking different but related topics. In general the questions are three.

Numerical Methods for eigenvectors and eigenvalues: general ideas. Localization theorems ad conditioning of the problem. Reduction to the Hessemberg form via orthogonal transformation. Givens and Householder matrices. Numerical methods for symmetric tridiagonal matrices. The Sturm method. The power method and its variants. The orthogonal iteration method and the QR method.

Singular value decomposition (SVD). Position of the problem. Existence and uniqueness of the SVD. SVD and low rank approximations.
SVD and condition number. SVD and pseudo inverse of a matrix. Rectangular linear systems: the solution of ordinary least squares problem via SVD.

Numerical methods for initial value problems. The Chauchy problem. General ideas on numerical methods. One step method. Runge Kutta methods and Runge-Kutta Fehlberg methods; Automatic algorithms for numerical integration.


Matlab for the solution of several problems in the mentioned domains.

D. Bini, M. Capovani, O. Menchi. Metodi Numerici per l'algebra lineare,Zanichelli, 1988.

M.G. Gasparo, Metodi Numerici per il cacolo di autovalori ed autovettori, valori singolari e vettori singolari di matrcii reali. Dispense

G. Monegato, Fondamenti di calcolo numerico, CLUT Editore, 1998.

A. Quateroni, R. Sacco, F. Saleri, Matematica Numerica, Springer, 2000.