ECTS - University of Pisa
| Modules | Area | Type | Hours | Teacher(s) | |
| METODI DI APPROSSIMAZIONE | MAT/08 | LEZIONI | 42 |
|
Il corso si concentra su tecniche avanzate di algebra lineare numerica e le sue applicazioni. Lo studente che completa il corso sarà a conoscenza di diversi argomenti avanzati di algebra lineare numerica, e avrà una descrizione dello stato della ricerca in questi campi.
Quest'anno il corso si concentrerà su funzioni di matrici ed equazioni matriciali tipo-Riccati (inclusa qualche applicazione alla teoria dei controlli), e (se il tempo lo permetterà) fasci e polinomi di matrici.
This course focuses on advanced techniques in numerical linear algebra (NLA) and their applications. The student who successfully completes the course will be aware of several advanced topics in numerical linear algebra. The student will be also able to have a look at the state of the local research in these fields.
This year the course will cover matrix functions, Riccati-type matrix equations (with some applications to control theory), and, time permitting, matrix pencils and polynomials.
(si veda la versione inglese del programma per informazioni dettagliate.)
The final exam is based on a seminar: the students will need to read, understand, and repeat the numerical experiments (to some extent) of a research paper in numerical linear algebra (to be agreed with the teacher), and make a short presentation on it.
Possibly, if the numbers allow it, a shorter lecture on a smaller topic will be assigned during the course.
(si veda la versione inglese del programma per informazioni dettagliate.)(si veda la versione inglese del programma per informazioni dettagliate.)
The student will be able to interface themselves with papers and research presentations in some areas of numerical linear algebra.
(si veda la versione inglese del programma per informazioni dettagliate.)
During the course and the final exam.
(si veda la versione inglese del programma per informazioni dettagliate.)
The material covered in undergraduate courses on numerical analysis and numerical linear algebra; a solid command of linear algebra and proof-based mathematics. Some experience with programming (and numerical programming), to implement and test the algorithms encountered. Working knowledge of the English language (if the course is taught in English).
(si veda la versione inglese del programma per informazioni dettagliate.)
Delivery: face to face
Learning activities:
Attendance: Not mandatory
Teaching methods:
(si veda la versione inglese del programma per informazioni dettagliate.)
(tentative) Kronecker products and Sylvester equations. Matrix pencils and their classification. Structure theorems for matrix pencils and polynomials. Some linearization methods for matrix polynomials. Introduction to matrix functions and equations. Numerical methods for generic and special matrix functions (exponential, sign function, etc.). Introduction to control theory. Lyapunov and Riccati equations: solvability properties and numerical methods (Newton, matrix sign iteration, SDA, and a brief outline of methods for sparse equations).
(si veda la versione inglese del programma per informazioni dettagliate.)
Higham, *Functions of matrices*.
Recommended reading includes excerpts from research monographies and papers published on journals in the field of computational mathematics and numerical/symbolic computing. Some material will be specified during the course.
A good research textbook to fill in on the standard algorithms is Golub, Van Loan, *Matrix computations*, 4th edition.
(si veda la versione inglese del programma per informazioni dettagliate.)
The lectures will be recorded (barring technical issues), so it is expected that even non-attending students can follow them. Non-attending students are expected to get familiar with the material covered in the course, and to contact the teacher to agree a topic for the final presentation.
(si veda la versione inglese del programma per informazioni dettagliate.)
Oral exam, including a presentation on a research topic decided beforehand in agreement with the teacher.
