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).

The course also contributes to the achievement of the following goals of the United Nations 2030 Agenda for Sustainable Development:

• Goal 4. Quality Education.
• Goal 5. Gender Equality.
• Goal 8. Decent Work and Economic Growth.

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

For the start date of the lectures, please refer to the following link.

https://corsi.unige.it/corsi/11894/studenti-orario

Class schedule

NUMERICAL ANALYSIS

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

The methods of exam preparation and the level of depth required for each topic will be specified during the lectures.

The written exam is intended to assess the effective acquisition of the fundamental knowledge of Numerical Analysis. The proposed exercises will evaluate the student’s ability to apply theoretical concepts to concrete problems, using numerical methods and techniques correctly.

The oral exam will cover the topics addressed during the lectures and aims to assess not only the mastery of the subject matter, but also the student’s ability to present mathematical concepts clearly and rigorously, using appropriate terminology. Evaluation will also take into account the understanding of definitions, theorem statements, and — where required — their proofs.

Students with disabilities or Specific Learning Disorders (SLD)

Students with certified disabilities, SLD, or other special educational needs are invited to contact the instructor by email within the first two weeks of classes in order to arrange personalized teaching and examination accommodations.
These accommodations, while respecting the learning objectives of the course, will take into account individual learning needs and may include appropriate compensatory tools where necessary.

For further details, please refer to the  section “Further Information”.

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

The request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it