OVERVIEW

Numerical Analysis is a fundamental pillar of Applied Mathematics, providing concrete tools for solving mathematical problems through the use of computers.

This course aims to equip students with numerical algorithms and mathematical tools to effectively and critically address and solve some of the most common problems in scientific contexts, including:

• the numerical solution of linear systems;
• the approximate computation of the zeros of a function;
• data interpolation using polynomials, particularly useful for processing experimental data.

Special attention is given to the stability of algorithms, the conditioning of problems, and the quality of approximations—key aspects for the informed use of computers in scientific applications.

The course provides practical skills that are valuable not only for laboratory work, but also for the interpretation and validation of numerical results in chemical practice.

AIMS AND CONTENT

LEARNING OUTCOMES

Knowledge and understanding of fundamental concepts and elements of numerical analysis. Particular emphasis is given to understanding the numerical aspects of problem solving, such as conditioning and stability, and to understanding the concept of approximate solution as a means of solving real problems.

AIMS AND LEARNING OUTCOMES

The Numerical Analysis course aims to provide students with fundamental knowledge and skills for addressing mathematical problems using computational methods, with particular attention to the needs of scientific and experimental practice.

By the end of the course, students will be able to:

•  understand the fundamental principles of numerical analysis and the role of approximation in scientific applications;
• apply basic numerical algorithms to solve elementary problems, such as the solution of linear systems, the computation of function zeros, and the interpolation of experimental data;
• evaluate the numerical stability and error sensitivity of the algorithms used;
• critically interpret numerical results, considering computational limitations and rounding errors.

The skills acquired will form a solid foundation for the quantitative analysis of data in future courses and laboratory activities.

In addition, through the use of quizzes on the AulaWeb platform, students will have developed personal competencies (such as self-awareness, focus, complexity management, critical thinking, decision-making, autonomy, and stress management) and the ability to "learn how to learn", including the organization and evaluation of their own learning process.

PREREQUISITES

To successfully follow the Numerical Analysis course, students are expected to have a solid understanding of basic mathematical concepts, typically acquired during secondary education.

In particular, students should be familiar with:

• solving equations and inequalities;
• fundamental concepts of analytic geometry;
• the concepts of function and derivative;
• reading and interpreting function graphs.

These skills are essential prerequisites for understanding and applying the numerical methods introduced during the course.

TEACHING METHODS

The Numerical Analysis course is worth 4 CFU (university credits), corresponding to 16 classroom lectures of 2 hours each, for a total of 32 hours. The lessons are held in person and in Italian.

Each lecture includes a theoretical introduction to fundamental concepts, followed by practical examples and exercises solved in class. This structure is designed to reinforce learning and develop the ability to apply numerical methods to real-world problems.

The learning process is further supported by self-study activities, including online quizzes available on the AulaWeb platform. These quizzes:

• can be repeated multiple times, offering students a flexible and interactive learning environment;
• help develop the ability to solve exercises independently within time constraints, thus fostering autonomy and responsibility;
• serve as an effective self-assessment tool, allowing students to continuously monitor their level of preparation.

This integrated approach—combining traditional lectures with digital resources—encourages active student participation and supports the development of transversal skills, such as the ability to learn how to learn.

SYLLABUS/CONTENT

The course covers the main topics of Numerical Analysis, with particular focus on computational and practical aspects. The program includes:

• Vectors and matrices: basic operations, properties, and notation.
• Vector and matrix norms: definitions and properties.
•  Linear systems: Gaussian elimination method, implementation, and properties.
• Condition number: analysis of the sensitivity of solutions with respect to the input data.
• Overdetermined systems: methods for approximate solutions, with particular reference to the least squares method.
• Regression line: introduction and computation of the linear regression line for data analysis.
• Errors, conditioning, and stability: classification of errors, problem conditioning, and stability of numerical algorithms.
• Root-finding for nonlinear equations: iterative methods (bisection, secant, and Newton’s method) and stopping criteria.
• Polynomial interpolation: construction of interpolating polynomials (Lagrange, Vandermonde).

RECOMMENDED READING/BIBLIOGRAPHY

The teaching materials presented during the course are explained and supplemented with examples in the slides prepared by the instructor, which are available on the AulaWeb platform.

To deepen the theoretical understanding and support independent study, the following textbook is recommended:

• Claudia Fassino, Introduzione al Calcolo Numerico, Società Editrice Esculapio

(available both in print and online at editrice-esculapio.com)

For preparing for the written exam and practicing with exercises similar to those on the test, the following workbook is also recommended:

• Claudia Fassino, Esercizi svolti di Calcolo Numerico, Società Editrice Esculapio

(available both in print and online at editrice-esculapio.com)

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

According to the timetable shown here

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written and an oral part. The writte part consists of exercises related to the theory, similar to the examples illustrated in class. The written test must be carried out before the oral exam and must be taken in the same session in which the student intends to take the oral exam.

To access the written exam, the student must have obtained a pass in all the quizzes on Aulaweb. This eligibility does not expire. To access the oral exam, students must have passed the written exam with a minimum mark of 18/30 and the mark obtained will be used in the final assessment.

There will be 2 exam sessions available for the winter session (mid-January-February) and 3 exam sessions for the summer session (June, July and September). Extraordinary exam sessions will not be granted outside the periods indicated in the study course regulations, with the exception of non-course students.

Students with DSA, disability or other special educational needs certification are advised to contact the teacher at the beginning of the course to agree on teaching and exam methods which, in compliance with the teaching objectives, take into account the learning methods individuals and provide suitable compensatory instruments.

ASSESSMENT METHODS

Details on how to prepare  the exam and on the degree of detail of each topic will be given during the lessons. The written exam will verify the effective acquisition of the basic knowledge of Numerical Calculus: the exercises will allow assessing the ability to apply theoretical concepts to specific problems. The oral exam will mainly focus on the topics covered during the lectures and will aim to evaluate not only if the student has reached an adequate level of knowledge, but if he/she has acquired the ability to explain mathematical concepts (definitions, theorems and proofs) in clearly and in correct terminology.

FURTHER INFORMATION

Students who have valid certification of physical or learning disabilities on file with the University and who wish to discuss possible accommodations or other circumstances regarding lectures, coursework and exams, should speak both with the instructor and with Professor Sergio Di Domizio (sergio.didomizio@unige.it), the Department’s disability liaison.