One of the main aims of the course is to provide the mathematical foundation underlying the different methods or algorithms, recall the main theoretical properties: stability, accuracy, algorithmic complexity, and show examples and counterexamples which illustrate the advantages and weaknesses. It also aims to test the algorithms presented in a simple and fairly universal software such as MATLAB.
Basics: Calculus and Linear Algebra
Course contents summary
Approximation of data and functions
- Numerical integration: Newton-Cotes formulas, formulas composed, formulas for integrals in multiple dimensions (Hint).
-Linear systems: direct methods, method eleiminazione Gauss factorization.
-Non linear equations.
-Introduction to Matlab.
Polynomial interpolation, Lagrange interpolation formula, Hermite interpolation formula, the formula of Newton divided differences, interpolation of piecewise polynomial functions, spline functions. Numerical integration: interpolatory quadrature formulas, according to Newton-Cotes Integration, Error estimates, Formule composed, Applications of quadrature formulas.
Numerical linear algebra: direct methods, the method of Gaussian elimination, Gauss decomposition and LU factorization. Equations and nonlinear systems: real roots of nonlinear equations, bisection method, secant methods, Newton-Raphson method.
G. Monegato, Fondamenti di Calcolo Numerico, CLUT.
Lectures and exercises in the classroom. MATLAB numerical exercises in the laboratory. Correction of exercises assigned individually