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
Error Analysis - Approximation of data and functions - Numerical integration: Newton-Cotes formulas - Hint formulas for integrals in multiple dimensions - Systems of linear equations: direct methods, factorization, iterative methods - Nonlinear equations - Introduction to Matlab
Error Analysis, Representation of numbers in a computer, rounding errors, machine
operations, Cancellation numerical conditioning of a problem and stability of an
Accuracy of data and functions: 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 NewtonCotes
Integration, Error estimates, Formule composed, Applications of quadrature
Numerical linear algebra: direct methods, the method of Gaussian elimination,
Gauss decomposition and LU factorization- Methods for Nonlinear equations -
A.Quarteroni, R.Sacco, F.Saleri: Matematica Numerica, Springer.
G.Naldi, Lorenzo Pareschi, G.Russo, Introduzione al Calcolo Scientifico, McGrawHill.
G.Monegato, Fondamenti di Calcolo Numerico, CLUT
Lectures and exercises in the classroom. MATLAB numerical exercises in the laboratory. Correction of exercises assigned individually.