B6217 - MATEMATICA NUMERICA 2

Course contents

Polynomial interpolation. Existence and uniqueness of the interpolation polynomial, Lagrange polynomial interpolation, the limitations of using equidistant nodes and the Runge phenomenon. Chebyshev polynomials. Study of interpolation error, stability, and convergence. Piecewise polynomial functions and splines.  

Numerical integration. Newton-Cotes interpolatory quadrature formulas (midpoint rule, trapezoid rule, Cavalieri-Simpson rule), composite and adaptive variants. Error analysis and convergence. 

Ordinary differential equations. Approximation of derivatives. Numerical methods for initial-value differential problems: one-step methods, higher-order methods (Runge-Kutta methods, and multistep methods). Analysis of methods: explicit/implicit methods, consistency, stability, convergence. Finite-difference method and finite-element method for boundary-value differential problems. Convergence analysis.

 

Throughout the course, examples from various application areas will be presented and discussed in relation to the different topics covered, and there will be laboratory exercises using the MATLAB scientific computing environment.

Readings/Bibliography

Teaching materials (slides, exercises, etc.) will be provided by the professors. These resources will be uploaded to the virtual platform as the course progresses.

Below is a list of recommended books:

- A. Quarteroni, R. Sacco, F. Saleri, Matematica Numerica, III ed., Springer-Verlag, 2008 e succ.

- A. Quarteroni, R. Sacco, F. Saleri, Matematica Numerica - Esercizi, Laboratori e Progetti, II ed., Springer-Verlag, 2013 e succ.

- A. Quarteroni, F. Saleri, P. Gervasio, Calcolo scientifico, VI ed., Springer-Verlag, 2017

- V. Comincioli, Analisi Numerica - metodi modelli applicazioni, McGraw-Hill, 1995.

Teaching methods

The course includes classroom lectures and hands-on laboratory work for developing programs corresponding to the algorithms covered, using the MATLAB scientific computing environment.

Considering the types of activities involved and the teaching methods adopted, participation in this training activity requires all students—including incoming international students, such as Erasmus students—to complete the General Training Course (Module 1) and the Low-Risk Training Course (Module 2) via E-Learning, available on the following webpage:

https://site.unibo.it/tutela-promozione-salute-sicurezza/en/training/students-general-and-specific-training

Assessment methods

The knowledge and skills to be acquired are assessed through a laboratory test and an oral exam.

The laboratory test consists of the presentation and discussion of various programs related to the algorithms covered in the course in MATLAB. This test is mandatory for admission to the subsequent oral exam, which is designed to assess the student's knowledge and understanding of the theory covered in the course.

Class attendance is very important in the learning process and strongly recommended, but it does not influence the evaluation process in any way.

To participate in the exam, it is necessary to register via AlmaEsami for the chosen exam session, by the scheduled deadlines.

Students with specific learning disorders (SLD) and/or temporary or permanent disabilities: please, contact the responsible University office (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students ) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the professors, who will assess the appropriateness of the adjustments, taking into account the teaching objectives of the course.

Teaching tools

Slides and other material provided in electronic format (exercise sheets, etc.).

Office hours

See the website of Silvia Tozza

See the website of Michele Ruggeri