ACADEMIC YEAR: 2014/2015
YEAR OF STUDY: 2
SEMESTER: Second semester
NUMBER OF CREDITS: 12
UNIT COORDINATOR: DILIGENTI Mauro
CONTACT HOURS: 104
INDIVIDUAL WORK HOURS: 104
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.
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 - Ordinary differential equations (discrete methods one step) - 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 algorithm.
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, interpolation of functions of several variables (hint).
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, Iterative Refinement, matrix inverse. Iterative methods: Jacobi method, Gauss-Seidel method, Method of sovrarilassamento (SOR).
Equations and nonlinear systems: real roots of nonlinear equations, bisection method, secant methods, the tangents (Newton-Raphson), Test of convergence, iterative methods in general, Aitken acceleration method.
Ordinary differential equations: methods one-step explicit Runge-Kutta methods, the local behavior of one-step methods, convergence of the methods one-step explicit estimate local truncation error and the choice of the integration step. Multistep methods (hint). Stability of numerical methods.
A.Quarteroni, R.Sacco, F.Saleri: Matematica Numerica, Springer.
G.Naldi, Lorenzo Pareschi, G.Russo, Introduzione al Calcolo Scientifico, McGraw-Hill.
G.Monegato, Fondamenti di Calcolo Numerico, CLUT.
Lectures and exercises in the classroom. MATLAB numerical exercises in the laboratory. Correction of exercises assigned individually.
Written test laboratory followed by an oral.