B5562 - MATEMATICA NUMERICA 2

Learning outcomes

At the end of the course, students complete the study of the fundamental problems in Scientific Computing, related to the approximation of functions, data, and integrals and the numerical treatment of ordinary differential equations. They can analyze the main numerical methods for efficiently solving these problems on a computer. Students can use the acquired knowledge to study and develop algorithms for solving problems arising in applications and they can critically analyze the obtained results in depth.

Course contents

Polynomial interpolation. Existence and uniqueness of the interpolation polynomial, Lagrange polynomial interpolation, Chebyshev polynomials, study of interpolation error, stability, and convergence. Piecewise polynomial functions and splines. Orthogonal polynomials.

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

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.

Given the type of activities and teaching methodologies adopted, participation in this training course requires prior attendance of all students in modules 1 and 2 of training on safety in the places of study, in e-learning mode https://elearning-sicurezza.unibo.it/

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