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 laboratory lessons, where numerical techniques are translated into MatLab Programs 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 air conditioning 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 complemented by practical lessons using Personal Computer

    Hours of lectures: 32 on 2 semesters, 2 hours per week (teacher Fassino)
    Lab hours: 24 on 2 semesters, 2 hours per week

(teacher Fassino with Federico Benvenuto )

The course also uses the didactic support provided by AulaWeb at the address

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

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 nuber of a matrix.

Overdetermined systems: the method of the normal equations. The regression line.
    
For the laboratory part, the use of the MatLab language.

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 three parts:

1) Laboratory test: a short MatLab program, written using the structures seen in Lesson. Rating from -2 to +2.

2) Written exam. It consists of two parts:

- a part concerning basic mathematics topics used during the course (max score: 5 points)

- a part composed of exercises concerning the whole theory developed during the classroom lessons (max score: 25). Rating in thirtieths.

3) Oral exam: questions concerning the theory carried out, with particular attention to theorems and proofs. Rating in thirtieths.

Final mark: average of the marks obtained in the written test and in the oral test to which, in an algebraic manner, the mark of the laboratory test is added. If this vote exceeds 30, praise will be given.

ASSESSMENT METHODS

The laboratory test aims 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.

The written test and the oral exam can be supported according to two modalities:

- 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 the written test and the oral exam concerning the program in the first semester and after the end of the second semester can be taken the written test and the oral exam concerning the program carried out in the second semester. The average of the marks reported in the two parts (together with the laboratory evaluation) gives the final mark.

Exam schedule