Teaching code

4S004793

Teacher

Elena Gaburro

Coordinator

Elena Gaburro

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/08 - NUMERICAL ANALYSIS

Period

Semester 1  dal Oct 1, 2024 al Jan 31, 2025.

Courses Single

Authorized

Lessons timetable Moodle Seminars 0

Learning objectives

The course will discuss, from both the analytic and computational points of view, the numerical solution of Mathematical problems such as: non linear systems, linear systems, matrix eigenvalues, interpolation and approximation, Gaussian quadrature. The objective therefore is to expand on the material introduced in Calcolo Numerico I and to introduce new and more sophisticated solution algorithms. In particular, we will present techniques that are fundamental for important modern problems of Applied Mathematics such as that of high dimensional datasets (SVD and PCoA) and optimization (conjugate gradient method). The course has a Laboratory component where the methods studied will be implemented using the MATLAB programming platform (using either the official Matlab from Mathworks or else the open source version GNU OCTAVE). At the end of the course the student will be expected to demonstrate that s/he has attained a level of competence in the computational and computer aspects of the course subject, as well as the ability to recognize which algorithms are appropriate for basic and advanced problems of numerical analysis.

Prerequisites and basic notions

Analysis 1, Linear algebra, differential calculus in one and several variables, integral calculus, basic methods of numerical calculus.
(It is possible to attend the course even without having passed all of the first-year exams, but it is necessary to keep in mind that the lecture modalities will not be adapted for any reason to those who have not passed the exams of Mathematical Analysis 1, Linear Algebra and elements of Geometry, Computer Programming witl lab, Physics 1 and Numerical analysis 1 with laboratory or equivalent exams.)

Program

The aim of this course is the analysis of numerical methods for the approximate solving of complex problems in applied mathematics.
The following topics will be covered in the course
- Methods for solving linear systems (classical iterative methods, conjugate gradient, QR and SVD factorizations, overdetermined systems)
- Methods for finding zeros of functions and systems (fixed point and Newton iterations for systems)
- Methods for finding eigenvalues and eigenvectors with application to google page ranking
- Interpolation methods
- Numerical quadrature methods (e.g. Gaussian formulas)
- Elements of machine learning (application of SVD)
- Optimization methods
The methods developed in the lecture will be further investigated, implemented on a computer and tested on various examples.
Note. The order of topics is subject to change.

Bibliography

Didactic methods

Theoretical lectures in the classroom and numerous lectures dedicated to the implementation, motivation and discussion of the numerical methods subject of the course.
MATLAB is required.

Learning assessment procedures

The examination consists of
A) a laboratory test in which the students will be asked to implement some programs in MATLAB, motivate them and comment on the obtained results
B) an oral examination of theoretical knowledge and skills
Test A is passed when a mark greater or equal to 18 is obtained.
You are only admitted to test B when you pass test A.

Evaluation criteria

The examination aims to ascertain that the student possesses knowledge and skills in the field of numerical analysis, numerical methods and their programming.

Criteria for the composition of the final grade

The grade will be obtained by taking the average of the laboratory test and the oral test.

Exam language

Italiano