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Degree course | Information and Communication Technologies (ICT) Engineering |

Curriculum | Curriculum unico |

Learnings | Orientamento unico |

Academic Year | 2016/2017 |

ECTS | 6 |

Scientific Disciplinary Sector | MAT/08 |

Year | Third year |

Time unit | Second semester |

Class hours | 48 |

Educational activity | Educational activities chosen by the student (art.10, paragraph 5, letter a) |

Supplying course | 50M006 CALCOLO NUMERICO E PROGRAMMAZIONE in Ingegneria Informatica e dei sistemi per le Telecomunicazioni LM-27 COTRONEI MARIANTONIA |

Professor | Mariantonia COTRONEI |

Objectives | Aim of the course is to provide the basic knowledge of the main methods in Numerical Calculus and to introduce Matlab and Octave as environments for scientific computing. The educational objectives include the acquisition of: ability to construct numerical models and to design algorithms for their solution; awareness of the problems connected to the use of a computer for solving mathematical problems; ability towards computer implementation of numerical algorithms on the computer, to realize numerical tests and to critically analyse the results. |

Programme | FLOATING-POINT ARITHMETIC AND ERROR ANALYSIS Computer representation of real numbers. Numerical accuracy. Floating-point arithmetic. Errors and error propagation. Conditioning of a mathematical problem. Algorithmic stability. SOLUTION OF NON-LINEAR EQUATIONS Iterative methods: convergence and convergence order. The bisection method. Newton-Raphson method: error analysis and error estimation. Secant method. Fixed point iteration. Stopping criteria. SOLUTION OF SYSTEMS OF LINEAR EQUATIONS Matrix arithmetic. Special matrices and their properties. Vector and matrix norms. Stability in solving linear systems. Condition number of a matrix. Direct methods. Solution of triangular systems. Gauss elimination. Pivoting. LU factorization. Cholesky factorization. Iterative methods. Iteration matrix. Convergence. Stopping criteria. Jacobi and Gauss-Seidel methods. Richardson methods. Gradient and preconditioned gradient methods. FUNCTION AND DATA APPROXIMATION Polynomial interpolation. Lagrange polynomial. Interpolation error. Runge phenomenon. Spline interpolation. Linear and cubic splines. Least square approximation. NUMERICAL DIFFERENTIATION AND INTEGRATION Approximation of derivatives. Finite differences. Interpolatory quadrature formulas. Order of exactness. Newton-Cotes and composite Newton-Cotes rules. Error formulas. Gauss-Legendre quadrature. Richardson extrapolation. SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Cauchy problem. One-step methods. Local and global errors. Consistency and convergence. Euler and Crank-Nicolson methods. Runge-Kutta methods. Multistep methods. AdamsBashforth and AdamsMoulton methods. Predictor-corrector schemes. NUMERICAL METHODS FOR BOUNDARY VALUE PROBLEMS Finite difference approximation of the Poisson problem in one and two dimensions. Finite element method. Finite element approximation of the Poisson problem in one dimension. AN INTRODUCTION TO MATLAB Main commands, arrays, built-in functions. Graphics. Design and development of programs. Implementation of numerical methods and analysis/validation of the results on test problems. |

Books | A. Quarteroni, F. Saleri, P. Gervasio. Calcolo Scientifico. Esercizi e problemi risolti con MATLAB e Octave, Springer, 2012. M.L. Lo Cascio, Fondamenti di Analisi Numerica, McGraw Hill, 2007. G. Naldi, L. Pareschi, Matlab: Concetti e Progetti, Apogeo, 2013. Quarteroni, R. Sacco, F. Saleri. Matematica Numerica, Springer, 2008 A. Quarteroni, F. Saleri, P. Gervasio. Scientific Computing with MATLAB and Octave, Springer |

Traditional teaching method | Yes |

Distance teaching method | No |

Mandatory attendance | No |

Written examination evaluation | No |

Oral examination evaluation | Yes |

Aptitude test evaluation | No |

Project evaluation | Yes |

Internship evaluation | No |

Evaluation in itinere | No |

Practice Test | Yes |

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