Teaching code

4S004793

Teacher

Elena Gaburro

Coordinator

Elena Gaburro

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/08 - NUMERICAL ANALYSIS

Period

1st semester dal Oct 1, 2025 al Jan 30, 2026.

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 and programming.
In particular, methods of direct type for solving linear systems, which are covered in the Numerical Calculus 1 course, will be explicitly required during this course.
(It is possible to attend the course even without having passed all 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

Classroom theory lectures:
i) explanations will be given using the blackboard (chalkboard if available, or otherwise a projected graphics tablet);
ii) the subject of the lectures will be the theoretical explanation of the topics and the demonstration of their properties;
iii) the reasons for the importance of the topics covered, the links and comparisons between related topics, comments on efficiency and possibilities of use will also be provided;
iv) possible applications will be explained;

Moreover, numerous laboratory lectures will be dedicated to the implementation, motivation, and discussion of the numerical methods covered in the course. For the laboratory lectures, the use of MATLAB software (version to be agreed upon with the professor) is required.

It is important to note that the course aims to develop a mindset capable of inventing, developing, mathematically motivating, structuring, and effectively implementing in a programming language, new and effective algorithms.
To this end, it starts from the theoretical study, motivations, and above all, the constructive implementation of well-known algorithms. The use of artificial intelligence is therefore NOT permitted because it would prevent students from learning the mechanism underlying the creation of algorithms and codes, completely replacing it in the case of simple algorithms.

Learning assessment procedures

The exam consists of two tests
A) a computer-based test that involves solving exercises within a given time, with the possibility of consulting personal material, but WITHOUT the possibility of accessing the internet, using any artificial intelligence tools, or communicating with classmates or external people.
The exercises involve implementing the solution in MATLAB, explaining and commenting on procedures and results.
B) a test of theoretical knowledge and skills structured as follows
b1) answer a theoretical question in written form;
b2) implement or modify, on the spot even from scratch, in front of the teacher, in MATLAB, algorithms covered in the course or solve short exercises on the computer (an insufficient result interrupts test B and may also CANCEL/modify the grade of test A).
b3) answer orally (with the aid of paper and pen if necessary) in a precise and coherent manner, without digressions, theoretical questions concerning the topics of the course, and be able to establish connections and comparisons between them; demonstrate skills that allow you to apply your knowledge to the resolution of questions and exercises.
Test A is considered passed when a grade greater than or equal to 18 is obtained. The grade is valid until it is MODIFIED/CANCELLED by question b2. You are only admitted to test B when test A has been passed.

Evaluation criteria

It will be verified that the student
- knows the theoretical concepts covered in the course and is able to establish relationships and comparisons between them.
- knows how to apply the knowledge acquired to both theoretical and practical exercises.
- can clearly explain concepts in written and oral form.
- can implement algorithms from scratch, modify and/or use existing algorithms.
- has developed a mindset suited to creating, developing, and implementing algorithms and codes.

Criteria for the composition of the final grade

Test A is considered passed when a score of 18 or higher, expressed in thirtieths, is obtained. The score is valid until it is MODIFIED/CANCELLED by question b2.
Students are only admitted to test B if they pass test A.
The final grade (between 18 and 30L) is obtained by calculating the average of the grade for test A (possibly modified by question b2) and test B. One point will be added to the final grade if the student has attended at least 75% of the lessons in person.

Exam language

Italiano