Module page
Analisi Numerica
| academic year: | 2013/2014 |
| instructor: | Silvia Noschese |
| degree course: | Mathematics - DM 270/04 (triennale), II year |
| type of training activity: | caratterizzante |
| credits: | 9 (72 class hours) |
| scientific sector: | MAT/08 Analisi numerica |
| teaching language: | italiano |
| period: | II sem (03/03/2014 - 13/06/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject:
- Linear and nonlinear systems of equations
Iterative methods for linear systems. Convergence and theorem of the spectral radius. Some iterative methods: Gauss-Seidel, Jacobi, relaxation. Stability of the algorithms. Conditioning analysis of the problem. Direct methods for some classes of matrices. Comparison of the methods. Newton methods in R^n. Nonlinear equations in the complex plane. Algebraic equations, numerical methods to compute all the roots. - Eigenvalues and eigenvector
Caracterization and localization of the eigenvalues, Gershgorin theorems. Global and individual conditioning numbers. Unitary transformations of similarity. QR factorization and QR method. Power method. Inverse iteration method. - Functions interpolation
Langrange and Hermite polynomial interpolation. Divided differences and Newton form. Orthogonal polynomials and Gaussian interpolation. Chebychev nodes and polynomials. Conditioning and stability of the algortihms. Lebesgue function and constant. Splines. Convergence properties. Minimum squared method. Numerical derivation via finite differences. Approximation of derivatives via splines functions. Richardson extrapolation. Sketch of the applications to partial differential equations. - Numerica integration
Newton-Cotes formulas. Gaussian formulas. Generalized quadrature formulas. Convergence and error estimates. Sketch on cubature formulas. - Ordinary differential equations
Numerical methods for the approximation of the Cauchy problem. Explicit and implicit schemes. One step and multistep methods. One step methods: consistency or order p, stability and convergence. Local and global error. Error analysis. Some one-step methods: Euler, Heun, modified Euler. Runge-Kutta methods. Adams formulas. Predictor-corrector schemes. Adams-Mooulton scheme. Error estimates. Sketch on the implementation of adaptive time step technique.
Detailed module subject: PROGRAMMA
Suggested reading:
A. Quarteroni, R. Sacco, F. Saleri, "Matematica Numerica", Springer, 2008
V. Comincioli, Analisi Numerica, Mc Graw Hill, 1990
Type of course: standard
Exercises:
- Foglio di esercizi n.1
- Foglio di esercizi n.2
- Foglio di esercizi n.3
- Foglio di esercizi n.4
- Foglio di esercizi n.5
- Foglio di esercizi n.6
- Foglio di esercizi n.7
Examination tests:
- Prima prova in itinere
- Seconda prova in itinere
- Prova scritta dell'appello del 25/06/2014
- Prova scritta dell'appello del 16/07/2014
- Prova scritta dell'appello del 08/09/2014
- Prova scritta dell'appello del 25/09/2014
- Prova scritta dell'appello del 26/01/2015
Useful link: registrazione prima prova in itinere
Prerequisites: Sono richieste nozioni di base di Analisi Matematica e di Algebra Lineare, quali quelle acquisite nei corsi di Calcolo I e II, Algebra Lineare e di Analisi Matematica I. E' inoltre richiesta la conoscenza di un linguaggio di programmazione (C, C++ o MATLAB) del livello acquisito nel corso di Laboratorio di Programmazione e Calcolo oppure in uno dei corsi di Abilita' Informatiche.
Knowledge and understanding:
The students will know the classical methods and algorithms of numerical analysis.
Skills and attributes:
The students will be able to choose a correct numerical method to solve their problem and they will be able to write the code corresponding to the algorithm.
Personal study: the percentage of personal study required by this course is the 65% of the total.