Learning objectives
Knowledge and critical use of numerical algorithms for the numerica solution of engineering applications.
Prerequisites
Knowledge and use of the basic elements of mathemathical analysis and linear algebra.
Course unit content
1. Interpolation and Approximation.
Polynomial interpolation. Newton's interpolation polynomial and divided differences.Interpolation at equally spaced points. Error of polynomial interpolation. Hermite's interpolation. polynomial. Interpolation by linear and cubic splines. Least-squares approximations. Notes on the Bézier curves. Applications.
2. Numerical Solution of Linear Systems.
Gauss elimination. Pivoting and scaling in Gauss elimination. Operations counts. Implementation of Gauss elimination and LU-decomposition. Iterative improvement. Vector norms. Matrix norms. Condition numbers and error estimates. Symmetric matrices (Cholesky method). Basic iterative methods (Jacobi's method, Gauss-Seidel method). Applications.
3. Solution of Nonlinear Equations.
Introduction. Bracketing methods. Newton's method. Rate of convergence. Applications.
4. Numerical Integration.
Introduction. Interpolatory numerical integration. Newton-Cotes formulas. Errors of quadrature formulas. Composite rules for numerical integration. Multiple integrals. Applications.
5. Ordinary Differential Equations.
Introduction. Cauchy value problem. Differential equations of the 1° order of normal form. Differential equations with separable variables. Homogeneous differential equations. Linear differential equations. Bernoulli equation. Riccati equation. Exact differential equations.Integration of some types of 2nd order differential equations.
6. Numerical Solution of Ordinary Differential Equations.
Introduction. One-step methods. Euler's method. Runge-Kutta methods. Implementation of variable-step Runge-Kutta.
7. Matlab
Full programme
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Bibliography
1. G. Monegato: Fondamenti di Calcolo Numerico. CLUT
2. G. Naldi, L. Pareschi, G. Russo: Introduzione al Calcolo SCIENTIFICO. MacGraw-Hill.
3. A. Mazzia: Laboratorio di Calcolo Numerico, Applicazioni con MATLAB e OCTAVE, Pearson.
4. L. Scuderi: Laboratorio di Calcolo Numerico, CLUT.
5. Lee W. Johnson, R. Dean Riess: Numerical Analysis, Addison- Wesley Publishing Company.
Teaching methods
Lectures, exercises and discussion on solving problems proposed by the teacher during the lectures.
Assessment methods and criteria
The exam consists of a short written test with some numerical exercises and programming of simple numerical algorithms in MATLAB and a subsequent oral test. It is possible to be exempted from the written test by delivering the exercises that will be proposed by the teacher during the period of the lessons. The exemption from the written test is valid for the entire academic year.
Other information
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