NUMERICAL ANALYSIS II
Learning outcomes of the course unit
A sound balancing of theoretical analysis, description of algorithms and discussion of applications is our primary concern.
Numerical Analisys 1, Numerical Laboratory.
Course contents summary
Approximation of Functions and Modelling of Data: Interpolation by linear and cubic splines. Trigonometric polynomials. Orthogonal polynomials and least-squares approximations. Least-squares fits.
Numerical Integration: Gaussian quadrature. Adaptive quadrature. Multiple integrals.
Solution of Linear Systems of Equations:QR-decomposition. Least-squares solution of overdetermined linear systems. Basic iterative methods. Jacobi method and Gauss-Seidel method. Implementation of iterative methods. Conjugate gradient Algorithm
Eigenvalue problem. Localization of eigenvalues. The power method. The inverse power method. Eigenvalues and eigenvectors of a Tridiagonal matrix. Reduction of a general matrix to Hessemberg form. Householder transformations. The QR Algorithm for real Hessemberg matrices.
Solution of Nonlinear Equations: Secant method, False Position method. Newton’s method in two variables. Zeros of polynomials. Fixed-point methods. Rate of convergence.
Numerical Solution of Ordinary Differential Equations: Linear multistep methods. Adams methods. Predictor-corrector methods. Order and convergence for multistep methods. Finite-difference methods. Collocation methods.
A. Quarteroni, R. Sacco, F. Saleri, Matematica Numerica, (1998), SPRINGER;
G. Naldi, L. Pareschi, G. Russo, Introduzione al Calcolo Scientifico (metodi e applicazioni con Matlab), (2001) McGraw-Hill
G.Monegato, Fondamenti di Calcolo Numerico, CLUT, Torino.