Errors and finite precision arithmetic; numerical methods for solving a nonlinear equation; Numerical solution of linear and nonlinear systems of equations; Approximation of functions and of definite integrals; Power method for eigenvalues and Google Pagerank.
L.Brugnano, C.Magherini, A.Sestini. Calcolo numerico, seconda edizione. Master, Universita' e Professioni, Firenze, 2010.
Learning Objectives
Conoscenze:
il corso si propone l'obiettivo di fornire gli strumenti di base di più comune utilizzo nel calcolo scientifico, con particolare enfasi sugli aspetti legati alla loro efficiente implementazione su calcolatore.
Competenze acquisite:
conoscenza dei metodi di base del calcolo scientifico.
Capacità acquisite al termine del corso:
capacita' di risolvere problemi di caclolo scientifico di base su calcolatore.
Prerequisites
Courses required: Analysis I: Integral and Differential Calculus, Linear Algebra, Programming.
Recommended Courses: Analysis I: Integral and Differential Calculus, Linear Algebra, Programming.
Teaching Methods
Credits: 9
Oral exam,
hours: 84
Further information
Partial examinations are reserved for the students attending the lectures.
Type of Assessment
Oral exam with report
intermediate for attending students
Course program
And finite arithmetic errors: errors of discretization, convergence errors, round-off errors, conditioning of a problem. The language Matlab.Radici of an equation: the bisection method, stopping criteria and conditioning of the problem, order of convergence, Newton's method, local convergence, the case of multiple roots, quasi-Newton methods. Solution of linear systems: simple cases, the LU factorization of a matrix, computational cost, diagonally dominant matrix, symmetric matrices and positive definite, LDL ^ T factorization, pivoting, conditioning of the problem, QR factorization and overdetermined linear systems. Basic iterative methods for solving linear systems: motivation, the Jacobi method, the Gauss-Seidel, splitting regular matrices. Outline of the basic methods for solving systems of nonlinear equations. Approximation of functions: polynomial interpolation, Lagrange shape and form of Newton interpolation error, conditioning of the problem, the Chebyshev abscissas, spline interpolation, cubic spline, polynomial approximation to the minimum quadrati.Formule quadrature: Newton-Cotes formulas , error and composite formulas, formulas adattative.Metodi for research of the eigenvalues of a matrix: the power method, applied to the calculation of "Google pagerank".