- Docente: Luca Ratti
- Credits: 6
- SSD: MAT/08
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: First cycle degree programme (L) in Chemical and Biochemical Engineering (cod. 8887)
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from Sep 17, 2024 to Dec 20, 2024
Learning outcomes
At the end of the course the student will have the ability to critically use specific computational methologies to solve general class of mathematical problems with application to process industry.
Course contents
Prerequisites:
Subjects of calculus and geometry from the courses Analisi Matematica T-1, T-2 e GEOMETRIA E ALGEBRA T. The course is delivered entirely in the Italian language.
Programme:
1) Introduction to numerical analysis.
Definition of numerical problems. Conditioning numbers.
Definition of algorithm, stability and computational complexity.
Finite-precision representation of real numbers. Sources of numerical error and its propagation.
2) Numerical methods to solve nonlinear equations.
Definition and theoretical study of the problem.
Methods: bisection, secants, Newton's, fixed point.
Convergence properties and efficiency.
3) Numerical resolution of linear systems.
Preliminaries: vector and matrix norms.
Definition and analysis of the conditioning of the problem.
Gaussian Elimintation and LU factorization algorithms, with and without pivoting.
Iterative methods (overview): gradient, conjugate gradient and extensions.
4) Numerical methods to solve nonlinear systems.
Methods: Newton's, Broyden's, fixed point.
Convergence properties and efficiency.
Examples and applications.
5) Numerical approximation of data and functions.
Polynomial interpolation: definition, theoretical properties.
Polynomial interpolation: numerical methods, conditioning, Runge's phenomenon.
Piecewise polynomial interpolation.
Least squares problems: definition, theoretical properties.
Least squares problems: QR factorization, iterative methods (an overview).
Readings/Bibliography
- A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, Matematica numerica, 4 ed., Springer, 2014.
- U.M Ascher, C. Greif, A First Course in Numerical Methods, SIAM, 2011.
- M.G. Gasparo, R. Morandi: Elementi di calcolo Numerico: metodi ed algoritmi, Mc-Graw Hill, 2008.
Teaching methods
Theoretical frontal lectures (3hrs per week)
- goal: acquire the key concepts and contents, prove the main theorerical results, discuss examples and exercises.
Exercise sessions in the lab (2-3hrs per week)
- goal: implement the main algorithms studied in class using Matlab, empirically verify their theoretical properties, tackle applications and examples.
Assessment methods
The final exam consists of a written test to verify
- the acquisition of theoretical contents (quiz);
- the development of problem-solving skills (resolution of exercises);
- the ability to implement the algorithms (coding or pseudo-coding exercises).
The final grade of the Integrated Course "ANALISI NUMERICA E LABORATORIO DI INFORMATICA T" is the weighted average (with weights 2/3 and 1/3) of the grades of this course, ANALISI NUMERICA T, and the one of the lab test of LABORATORIO di INFORMATICA T.
Teaching tools
Virtuale platform for additional material.
Office hours
See the website of Luca Ratti