OVERVIEW

The course covers the fundamental topics of Numerical Analysis, with particular emphasis on error analysis, numerical linear algebra, and the numerical solution of ordinary differential equations.

The theoretical content is complemented by practical laboratory sessions, conducted with the aid of a computer, aimed at the hands-on implementation of the algorithms introduced during lectures.

Both the theoretical knowledge acquired in the classroom and the practical skills developed in the lab contribute to the student’s final evaluation.

Lectures are held in Italian.

AIMS AND CONTENT

LEARNING OUTCOMES

The teaching aims to offer mathematical and methodological notions that point out some basic techniques of scientific computation. Integral part of the course are the laboratory exercises where the student apply the notions learned in class.

AIMS AND LEARNING OUTCOMES

The course aims to develop in students a mathematically grounded, application-oriented approach to problem solving, inspired by real-world phenomena. Particular attention is given to the treatment of data affected by errors and to the analysis of numerical methods for solution approximation.
The course is designed to provide conceptual and practical tools to:

• understand and model phenomena through applied mathematics;

• analyze the efficiency and stability of numerical methods;

• critically interpret the results produced by computational tools.

The course includes laboratory activities, during which students will develop practical skills in algorithm implementation, results analysis, and team collaboration.

At the end of the course, students will be able to demonstrate the following learning outcomes:

Knowledge and Understanding

• Understand the fundamental principles of numerical analysis and the main approximation methods.

• Identify sources of error in experimental data and numerical computations.

Applying Knowledge and Understanding

• Apply numerical methods to solve mathematical problems such as: linear systems, eigenvalue computation, and ordinary differential equations.

• Use computational tools to implement algorithms and critically analyze the outcomes.

Making Judgements

• Evaluate the quality and reliability of numerical results, recognizing the main sources of error.

Communication Skills

• Clearly and rigorously communicate the procedures followed and the results obtained, including in collaborative settings.

Learning Skills

• Develop both autonomous and collaborative working methods, especially through group work carried out in the laboratory

PREREQUISITES

Students are expected to have a solid understanding of fundamental concepts in mathematical analysis, including:

• continuity and differentiability of real-valued functions of a real variable,

• Taylor series expansions,

• ordinary differential equations.

Basic knowledge of linear algebra is also required, in particular:

• operations with vectors and matrices, and the solution of linear systems,

• concepts related to eigenvalues and eigenvectors.

Finally, a basic knowledge of programming techniques is necessary, as it is essential for the implementation and testing of the algorithms studied in the course.

TEACHING METHODS

Lectures are held in person, in the classroom, and conducted in Italian. During these sessions, students acquire fundamental knowledge and concepts of Numerical Analysis and practice problem-solving skills. An online quiz system on Aulaweb is provided to facilitate content comprehension and enable self-assessment.

Laboratory Exercises (2 ECTS, 24 hours, 2nd semester)

Laboratory activities deepen topics related to the theoretical lectures and are carried out in groups. Students implement code to solve assigned exercises. The produced material will be used during the laboratory exam. Attendance is mandatory for at least 80% of the sessions, except in special cases (e.g., working students).

 Final Evaluation

The final exam grade is a weighted average of the oral exam (dominant) and the laboratory evaluation. Passing requires achieving a passing grade in both components.

SYLLABUS/CONTENT

Lectures

The course covers the following fundamental topics in numerical analysis:

• Error theory: analysis of problem conditioning and numerical method stability.

• Numerical solution of linear systems: conditioning of linear systems; Gaussian elimination with pivoting strategy; LU and QR factorizations.

• Eigenvalue computation: power method and its variants; similarity transformations; introduction to the QR method.

• Singular Value Decomposition (SVD): existence and uniqueness of the decomposition, fundamental properties, and applications in numerical approximation.

• Function approximation: discrete least squares method, solved via normal equations and SVD.

• Ordinary Differential Equations (ODEs): one-step methods (e.g., Euler, Runge-Kutta) and multistep methods.

Laboratory Activities

Laboratory sessions are conducted in groups with the use of computers and include:

• Guided practical exercises on topics covered in the theoretical lectures.

• Implementation of numerical algorithms using a programming language (e.g., Python).

• Critical discussion of the results obtained, with particular focus on error analysis and the effectiveness of the methods used.

RECOMMENDED READING/BIBLIOGRAPHY

Lecture notes, written by Fassino and Piana, available on AulaWeb.

Book: Bini, Capovani, Menchi: “Metodi Numerici per l’Algebra Lineare". Ed. Zanichelli

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Final Exam. The final exam consists of two parts:

Oral part: This involves the presentation and discussion of selected topics, chosen by the instructor, from those covered during the theoretical lectures. Access to the oral exam is granted only to students who have passed the laboratory part.

Laboratory part: Conducted individually in the computer lab, this involves using some of the programs implemented during the course. The exam focuses on interpreting the obtained results and analyzing specific problems assigned during the exam.

Evaluation and Passing Criteria

• The oral exam grade can reach a maximum of 27/30, with a passing grade set at 15/30.

• The laboratory exam grade can reach a maximum of 5/30, with a passing grade set at 3/30.

Passing the exam requires achieving a passing grade in both parts. The final grade is the sum of the two parts. A final score of 31 or 32 is awarded the maximum grade of 30 cum laude.

Exam Sessions

Two exam sessions will be available during the winter session (January and February) and three during the summer session (June, July, and September). Extraordinary exam sessions outside the official periods set by the Study Program regulations will not be granted, except for students who are beyond the regular course duration.

Students with Disabilities and Special Educational Needs

Students with certified Specific Learning Disabilities (DSA), disabilities, or other special educational needs are advised to contact the instructor via email within the first two weeks of the course to arrange suitable teaching and exam accommodations. These accommodations, while respecting the course objectives, will consider individual learning needs and provide appropriate compensatory tools.

Students with disabilities or Specific Learning Disorders (DSA) are reminded that to request exam adaptations, they must follow the detailed instructions available on Aulaweb:
https://2023.aulaweb.unige.it/course/view.php?id=12490#section-3
In particular, accommodations must be requested well in advance (at least 10 days before the exam date) by writing to the instructor, copying the School’s Disability Representative and the competent office (see instructions).

ASSESSMENT METHODS

Assessment

The assessment consists of two parts:

Laboratory test: Conducted individually, it aims to verify the student’s ability to correctly implement and apply the numerical methods presented during the lectures. The correctness and efficiency of the produced code, as well as the ability to critically interpret the obtained results, will also be evaluated.

Oral exam: This involves the discussion of the main theoretical topics covered in the course, with particular attention to the understanding of definitions, statements, and proofs of theorems, as well as the ability to present them clearly, rigorously, and using appropriate mathematical language.

Passing the exam requires achieving a passing grade in both parts.

Guidelines regarding exam preparation and the required level of depth for each topic will be provided during the lectures.

FURTHER INFORMATION

Prerequisites

Basic knowledge of:

• Mathematical Analysis: functions, derivatives, introduction to differential equations.

• Linear Algebra: matrices, vectors, solving linear systems.

• Programming techniques at an introductory level.

Attendance Policy

• Theoretical lectures: attendance recommended.

• Laboratory: attendance mandatory for at least 80% of the hours, except in cases of documented impossibility.

Exam Registration

Exam registration must be completed exclusively online via the university portal, at least 5 days (120 hours) before the exam date.