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.

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.