OVERVIEW

The course resumes some of the topics that were introduced in "Fondamenti di Calcolo Numerico", and introduces new ones, with the aim of illustrating fundamental themes that might be encountered in the applications

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of this teaching is to introduce mathematical techniques borrowed from different fields such as analysis, geometry and algebra, and use them to solve mathematical problems originating in the applications. The course also envisages laboratory classes, where students will implement some of the techniques using Matlab.

AIMS AND LEARNING OUTCOMES

At the end of this course, you will:

• know the fundamental numerical techniques for solving linear systems iteratively;
• understand converge issues and error control in iterative methods;
• know the fundamental numerical techniques for solving interpolation and integration problems;
• understand the relationships between the different topics and the different techniques addressed in the course;
• be capable of implementing the numerical techniques.

PREREQUISITES

Basic knowledge in the following fields will is required for a good understanding of the classes: vector spaces and norms; function spaces; sequences and convergence; random variables and law of large numbers.

TEACHING METHODS

Frontal classes and laboratory exercises, aimed at implementing the tecniques presented in class and/or using them to solve applied problems.

SYLLABUS/CONTENT

• Methods for the solution of nonlinear equations.
• Iterative methods for the solution of linear systems.
• Minimization of quadratic forms: gradient and conjugate gradient method. 
• Polynomial interpolation.
• Brief introduction to Fourier series and the Discrete Fourier Transform
• Spline and trigonometric interpolation.
• Least squares.
• Numerical integration: Newton-cotes quadrature rules.
• Composite quadrature formulae: trapezoidal rule and Cavalieri-Simpson rule.
• Orthogonal polynomials and Gaussian quadrature.
• Brief introduction to Monte Carlo integration.

RECOMMENDED READING/BIBLIOGRAPHY

- G. Monegato - Fondamenti di Calcolo Numerico - CLUT 1998
- D. Bini, M. Capovani, O. Menchi - Metodi Numerici per l' Algebra Lineare - Zanichelli 1988
- R. Bevilacqua, D. Bini, M. Capovani, O. Menchi - Metodi Numerici - Zanichelli 1992.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral exam, assessing both knowledge of the theoretical part and understanding of the laboratory classes.

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.

Subsequently, significantly in advance (at least 7 days) of the examination date, an e-mail must be sent to the teacher with whom the examination will be taken, including in copy knowledge of both the School Contact Person for the Inclusion of Students with Disabilities and DSA (sergio.didomizio@unige.it) and the above-mentioned Sector. The e-mail must specify

- the name of the teaching course

- the date of the call

- the student's surname, first name and roll number

- the compensatory tools and dispensatory measures considered functional and required.

The contact person will confirm to the teacher that the applicant has the right to request adaptations during the examination and that these adaptations must be agreed upon with the teacher. The lecturer will respond by stating whether the requested adaptations can be used.

Requests must be sent at least 7 days before the date of the exam in order to allow the lecturer to assess their content. In particular, if concept maps are to be used for the exam (which must be much more concise than the maps used for study) if the submission does not meet the deadline, there will be no technical time to make any changes.

For further information on requesting services and adaptations, please consult the document: Guidelines for requesting services,

ASSESSMENT METHODS

At the oral exam, the candidate might be required to:

• introduce a general topic, such as "Lagrange interpolation" or "root finding algorithms";
• prove one of the main results presented and proved during classes;
• represent a problem graphically;
• discuss one of the Matlab codes that were produced during the laboratory classes.