Subject: NUMERICAL CALCULUS (A.A. 2023/2024)
Unit Calcolo numerico
The course of Numerical Analysis aims to train students able to analyze from a numerical point of view basic mathematical problems (such as linear algebra, roots finding or data/functions approximation), selecting the most appropriate algorithm according to the peculiarities of the specific problem. The implementation of the analyzed methods in Matlab language will allow the student on one hand to put into practice the theoretical knowledge and on the other hand to acquire knowledge and practice of a programming language that leads the world in scientific computing.
- Differential calculus for real functions of real variables.
- Integral calculus for real functions of one real variable.
- Basics of linear algebra.
- Basics of computer programming.
Computer representation of numbers: rounding errors and floating-point arithmetic.
Systems of linear equations: stability analysis, Gaussian elimination and matrix decompositions (LU, Crout, Cholesky, QR and SVD); iterative methods (Jacobi and Gauss-Seidel), convergence analysis.
Nonlinear equations and systems of nonlinear equations: the bisection method, the Newton method, secant method, fixed point iterations, convergence analysis, stopping rules.
Data and functions approximation: basic functions for approximation, polynomial interpolation, interpolation by splines, least-squares approximation.
Numerical integration: interpolatory numerical integration, Newton-Cotes formulas, errors of quadrature formulas, composite rules for numerical integration.
Eigenvalues computation: power method, inverse power method, iterative QR.
Introduction to computer programming in the MATLAB environment, implementation of numerical algorithms.
- Lectures in the classroom, with illustration of the content chapters by means of slides and blackboard. - Laboratory exercises on the numerical solution of the problems described during the lectures, for the practical verification of structured concepts that are the backbone of the course program.
Examination: oral exam. The candidate must demonstrate a thorough knowledge of: - the course content and teaching training, including both the institutional part, i.e. frontal lessons/classroom activities, and the laboratory practices; - the memorization and the operations of and between real numbers on the computer; - the main techniques for the solution of a linear system or a nonlinear equation, and the computation of eigenvalues and eigenvectors of a matrix; - the main notions on the data fitting problem, as the difference between interpolation and regression, the numerical strategies to design and compute an optimal model and the application to the numerical computation of the integral of a function; - the MATLAB syntax for the implementation of an elementar algorithm. The verification is integral with respect to the course's content; it is also verified the student's ability to relate specific subject content with the knowledge listed as pre-requisite. The oral exam consists in the implementation of one of the algorithms analyzed during the class in a Matlab environment and in the in-depth analysis of some topics treated during the lectures. The score of the oral exam, in a scale of thirty, is divided in: 5 points for the communicative skills; 5 points for multidisciplinary skills; 20 points for the knowledge of the contents.
KNOWLEDGE AND COMPREHENSION
- knowledge of the basic techniques for the solution of a linear system, the computation of eigenvalues and eigenvectors of a matrix, the solution of a nonlinear equation, the computation of a continuous model approximating a set of points and the estimate of the integral of a real function.
CAPACITY TO APPLY KNOWLEDGE
- solve numerically elementary mathematical problems choosing and applying the most suited mathematical technique;
- capability to effectively and efficiently implement an algorithm in the MATLAB language.
CAPABILITY TO JUDGE
- Capability to individuate among different algorithms for the numerical solution of a given mathematical problem according to the peculiarities and the structure of the data at disposal.
- Discussion in classroom and preparation of final exam give the student communication capabilities and relational skills to argument the obtained results or discuss results presented by others.
Through the overall course activities the student:
- acquires capability to deepen using bibliography the acquired knowledge;
- is capable of retriving a suitable procedure by the analysis of appropriate sources on the web or reference textbooks.