Learning outcomes of the course unit
- Knowledge and ability to understand the language and the typical problems in the transition from continuous mathematics to discrete mathematics.
- Ability to apply knowledge and understanding in critical analysis of obtained numerical results.
- Autonomy of judgment in evaluating the approximation algorithms and the obtained results also through discussion with one's peers.
- Ability to clearly communicate the concepts acquired and to argue the results achieved.
- Ability to learn limits and advantages of numerical methods and to apply them consistently.
One of the main aims of the course is to provide the mathematical foundation underlying the different methods or algorithms, recall the main theoretical properties and show examples and counterexamples whic illustrate the advantages. It also aims to test the algorithms presented in a simple and fairly universal software such as Matlab.
Basic concepts of Mathematical Analysis and Linear Algebra.
Basic: calculus and Linear Algebra.
Course contents summary
- Introduction to MATLAB.
- Error analysis.
- Approximation of data and functions.
- Numerical integration by Newton-Cotes formulas.
- Resolution of linear systems: direct methods, factorizations, iterative methods.
- Numerical resolution of non-linear equations.
Approximation of data and functions (14 hours)- Numerical integration: Newton-Cotes formulas (8 hours) - Systems of linear equations: direct methods, factorization (14 hours) - Non-linear equations (6 hours) - Introduction to Matlab (4 hours)
Accuracy of data and functions: polynomial interpolation, Lagrange interpolation formula, Hermite interpolation formula, the formula of Newton divided differences, interpolation of piecewise polynomial functions, spline functions, interpolation of functions of several variables (hint).
Numerical integration: interpolatory quadrature formulas, according to Newton-Cotes Integration, Error estimates, Formule composed.
Numerical linear algebra: direct methods, the method of Gaussian elimination, Gauss decomposition and LU factorization, matrix inverse. Non linear equations: bisection method.
- "Numerical analysis". L.W. Johnson, R.D. Riess. Addison-Wesley (1982).
G.Naldi, L. Pareschi, G. Russo, Introduzione al Calcolo Scientifico, McGraw-Hill.
G. Monegato, Fondamenti di Calcolo Numerico, CLUT.
After an initial introduction to the Matlab programming language, the course contents will be analyzed highlighting the problems related to the introduced numerical techniques. The course will also provide a part of re-elaboration in cooperative learning, supervised by the professor, consisting in the application of numerical techniques, through
programming in Matlab. This activity will allow the student to acquire the ability to face "numerical" difficulties and to evaluate the reliability and consistency of the obtained results.
Lessons and exercises in the classroom. Delivery of exercises assigned individually and/or to groups to be developed and resolved.
Assessment methods and criteria
The exam includes a written test regarding knowledge and skills acquired during the course. The threshold of sufficiency is fixed to the knowledge of the algorithms proposed during the course and to their implementation
in the Matlab language.
Written exam with exercises similar to those assigned in the exercises. Oral exam.
You are exempted from the written test if the proposed exercises are delivered during the course.