LEARNING OUTCOMES

Knowledge and understanding of concepts and fundamentals of numerical computation.
Particular emphasis is given to:
• the understanding of the aspects related to the numerical solution of problems such as conditioning and stability;
• the understanding of the concept of approximate solution as a means to solve real problems.

Knowledge and understanding of concepts and fundamentals of numerical calculation. Particular emphasis is given to the understanding of numerical aspects related to the solution of problems, such as the condition number of a matrix and stability; the understanding of the concept of approximate solution as a means to solve real problems.

AIMS AND LEARNING OUTCOMES

The main purpose of the teaching of Numerical Calculus and Programming is to provide tools for the calculation or approximation of the solution of basic mathematical problems.

The main objective is to shift the point of view, in dealing with mathematical problems, from a completely abstract field to a more applied one, in order to prepare the student to deal with problems deriving from the study of real phenomena.

Particular attention is paid to basic numerical concepts, such as the conditioning of a problem and the stability of an algorithm, and to the critical interpretation of the results.

Specifically, the student will be able:
- to solve linear systems of any size;
- to calculate the regression line associated with a set of points;
- to analyze errors due to perturbed data and / or floating point arithmetic;
- to approximate the zeros of a function;
- to find the polynomial that interpolates a set of  points.

PREREQUISITES

The basic concepts of analysis, analytical geometry and trigonometry taught in high school.

TEACHING METHODS

Theoretical lessons: 4 CFU (32 over two semesters). These lessons are provided both for students of "Scienza dei Materiali"  and for students of "Chimica e Tecnologie Chimiche".

At the end of each lesson, summary slides are published on Aulaweb.

Quizzes on aulaweb allow each student to self-evaluate.

Exercises. For students of Materials Science an additional 2 CFU are awarded (16 hours) (1 for each semester).

During the first semester, by applying the innovative teaching technique "Team Based Learning" (TBL), specific numerical problems are analyzed and solved.

During the second semester an interdisciplinary exercise is proposed, to be solved using notions of Chemistry, Physics and Mathematics.

Lectures and exercises are held in-person. Should the health emergency persist, to safeguard everyone's health and safety, the lessons will be delivered online on the Teams platform (and recorded).

SYLLABUS/CONTENT

The program covers topics from different areas:

• Matrix operations, vector and matrix norms. 

• Solution of linear systems: backward substitution method for triangular systems, the Gauss method.

• The condition number of a matrix.

• Overdetermined systems: the method of the normal equations. The regression line.

• Error Analysis: the use of the floating point arithmetic and algorithmic errors, the cancellation and round-off error. Conditioning of the problem of in the evaluating a real function.

• Solution of nonlinear equations: the bisection method, the Newton-like methods.

• Interpolation: the interpolating polynomial in the Lagrange form.
  

RECOMMENDED READING/BIBLIOGRAPHY

Bevilacqua-Bini-Capovani-Menchi: “Introduzione alla Matematica Computazionale”, Zanichelli
Bini-Capovani-Menchi: “Metodi Numerici per l’Algebra Lineare”, Zanichelli

Handouts provided by the teacher and available on AulaWeb at the address

https://smfc.aulaweb.unige.it/course/view.php?id=1047