OVERVIEW

The course is an introduction to the Numerical Analysis, and it consists in the description of strategies and algorithms for basic mathematical problems solution. Particular attention is  paid to the use of computer and to the analysis  of the numerical problems linked to it, as the error analysis  or the computational complexity. Complete the course some guided exercises, carried out in groups, during which numerical techniques are applied for solving simple mathematical problems, with particular attention to the interpretation of the numarical results.

AIMS AND CONTENT

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

Resolution, from the numerical point of view, of basic mathematical problems, such as the root finding of a real function, polynomial interpolation, the solution of square linear systems or of overdetermined linear systems, using direct or iterative methods.

Particular attention is paid to some basic numerical concepts, such as the conditioning of a problem and the stability of an algorithm,  and to the interpretation of the results obtained by using floating point arithmetic.

The main objective is to move the point of view of the students, in dealing with mathematical problems, from a completely abstract sphere to a more applied one, in order to prepare them to solve problems deriving from the study of real phenomena.

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".

Due to the health emergency and the high number of students (around 150), to safeguard the health and safety of all, the theoretical lessons will be  online on the Teams platform (at least during the first semester).

The lessons, provided synchronously (live), are also recorded so that each student can relate to the lessons, whenever he deems it appropriate.

The slides used are published, at the end of each lesson, on Aulaweb.

Innovative teaching techniques such as:

    quiz on aulaweb
    glossary on aulaweb

allow a better understanding of the course.

Exercises and insights. Additional 2 CFU (16 hours) are provided for students of "Scienza dei Materiali" (1 for each semester) during which, applying the innovative teaching technique "Team Based Learning" (TBL), specific problems are analyzed and solved. This activity will be carried out in  presence, if the health situation allows it (given the lesser number of students involved), otherwise it will be carried out on the Teams platform.

SYLLABUS/CONTENT

The program covers topics from different areas:

Error Analysis: the use of thefloating 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, analysys of the interpolation error.

Matrix operations, vector and matrix norms.

Solution of linear systems: backward substitution method for triangular systems, the Gauss method and Jacobi method for square systems.
    
The condition number of a matrix.

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

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

TEACHERS AND EXAM BOARD

Exam Board

CLAUDIA FASSINO (President)

FEDERICO BENVENUTO

LESSONS

LESSONS START

The course is developed on the first and second semester, following the timetable set out in the "Manifesto"

Class schedule

NUMERICAL CALCULATION AND PROGRAMMING

EXAMS

EXAM DESCRIPTION

The exam consists of two parts, with different characteristics, based on the health emergency.

1) Written exam: exercises concerning the whole theory carried out during classroom lessons. (in presence, if possible, or replaced by Quiz on aulaweb)

2) Oral exam: questions regarding the theory carried out, with particular attention to theorems and proofs (in presence, if possible, or on the Teams platform)

Final mark: it is given by the average of the written mark, of the oral mark and, for the students of Materials Science, of the mark reported in the group work (TBL).

The written test and the oral test can be taken in two ways:

- at the end of the course, and therefore cover the entire program

- or the exam can be divided into two parts. After the end of the first semester you can take the written test and the oral test concerning the program carried out in the first semester and after the end of the second semester you can take the written test and the oral test concerning the program carried out in the second semester. The average of the marks reported in the two parts gives the final mark.

ASSESSMENT METHODS

The guided exercises aim to verify the ability to solve, from a numerical point of view, simple mathematical problems.

The written test is based on the solution of exercises related to the theory carried out in the classroom, to ascertain the ability to analyze and solve a numerical problem.

The oral test aims to verify the understanding of the theory part, with particular attention to the proof of the theorems.

Exam schedule