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

LEARNING OUTCOMES

The aim of this course 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 in the C language, within a Matlab environment.

AIMS AND LEARNING OUTCOMES

At the end of this course, the student 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 in C language within a Matlab environment.

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.

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.
• Spline and trigonometric interpolation.
• Least squares, the continuous case.
• 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.

Teaching methods

Frontal classes and laboratory exercises.

LESSONS START

The class will start according to the academic calendar.

ORARI

L'orario di tutti gli insegnamenti è consultabile su EasyAcademy.

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NUMERICAL ANALYSIS

Exam description

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