# Course Details for A.Y. 2013/2014

#### Name:

**Analisi Numerica / Numerical Analysis**

### Basic information

##### Credits:

*:* Laurea in Matematica 9 CFU (b)

##### Degree(s):

Laurea in Matematica 2° anno curriculum Generale Obbligatorio

Laurea Magistrale in Matematica 2° anno curriculum Generale Obbligatorio

##### Language:

Italian

### Course Objectives

Goals of the course:
Provide the mathematical instruments ncessary to the numerical solution of basic problems in applied sciences,
to the development of algorithms in a structured programming language (like Matlab). The course consists of 9
credits which correspond to 90 hours.
Expected learning results:
Being able to solve numerically basic problems of analysis and linear algebra in the framework of simple mathematical
models in applied sciences and develope ability in the development of codes.

### Course Content

- Error analysis:
Representation of real numbers in a certain base. Floating point representation.
Machine number. Truncation and rounding. Machine operations. Error propagation.
Condition number of a problem. Stability of an algorithm.
Elements of programming. Matlab structured language.
- Numerical linear algebra:
Vectors, matrices and their properties. Norms. Eigenvalues and eigenvectors. Spectral radius.
Relationships between norms and spectral radius.
Special classes of matrices (Hermitian matrices, positive definite matrices etc.)
Direct methods for the solution of linear systems. Triangular systems, Gaussian elimination,
pivoting. LU and LLH factorizations,
Choleski factorization. Condition number of a linear system.
- Elements of programming:
Implementation of algorithms in MATLAB.
Laboratory sessions: Implementation of LU and Cholesky factorizations.
- Eigenvalue and eigenvector computation.
Localization of eigenvalues in the complex plane.
Perturbation theory for eigenvalues.
Power method and Wielandt variant for the computation
of eigenvalues and eigenvectors.
Laboratory sessions (2): approximation of eigenvalues by means of
power and inverse power methods.
- Interpolation and approximation. Computation of an algebraic polynomial
at a given point. Polynomial interpolation. Lagrange representation.
Linear interpolation operator. Interpolation error.
Chebyshev polynomials: recursive formula, zeros, minimal norm property.
Computation of an interpolating polynomial. Devided differences and Newton formula.
- Convergence of interpolatory schemws. Piecewise polynomial interpolation,
Spline functions. Computation of spline functions. Cubic splines.
Laboratory sessions (2): uniform polynomial interpolation and Chebyshev interpolation.
- Quadrature formulas: General form. Order and polynomial order.
Interpolatory formulas. Convergence theorem. Newton-Cotes formulas.
Gaussian formulas, Euristic error estimate. Composite formulas:
trapezoidal and Simpson rule. Adaptive quadrature.
Laboratory sessions (2): development of an adaptive integrator.
- Iterative methods for large linear systems. Splitting methods; general
convergence theorem; error control; Jacobi and Gauss-Seidel methods;
convergence of the methods on diagonally and weakly diagonally dominant
systems.
Laboratory session: application of iterative methods,

### Prerequisites and Learning Activities

Basic knowledge of calculus, geometry and linear algebra.

### Assessment Methods and Criteria

Written examination with exercises and theoretical tests. Written examination for the laboratory part
with development of algorithms in Matlab language. Optional oral examination.

### Textbooks

- J.Stoer, R.Bulirsch. , Introduction to Numerical Analysis , Springer Verlag. 1993.
* *
- A. Quarteroni, R. Sacco, F. Saleri. , Matematica Numerica , Springer - Collana Unitext. 2000.
* *
- D.Bini, M.Capovani, O.Menchi., Metodi numerici per l'algebra lineare , Zanichelli. 1988.
* *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 07 febbraio 2014, 17:59*