NUMERICAL METHODS FOR APPLICATIONS
Learning outcomes of the course unit
One of aims of this course is to present background material of several numerical methods in branches of mathematics with which numerical analysis has made its principal contacts.
First principles of Analysis and Geometry.
Course contents summary
Stability and Conditioning. Interpolation and Approximation. Polynomial interpolation. Hermite interpolation. Runge phenomenon for standard polynomial interpolation. Splines. Least-square approximations. Trigonometric interpolation.
Solution of linear system of equations. Gauss elimination. Operations counts. Pivoting and Scaling in Gauss elimination. LU decompositions. Cholesky decomposition. QR decomposition. Least-squares solution of over-determined linear systems.
Numerical integration. Newton-Cotes Formulas. Composite formulas. Error estimate. Adaptive quadrature.
Solution non-linear equations. Bisection. Newton method in one variable. Secant method. Construction. Practical considerations.
Ordinary Differential Equations. Initial-value problems. One-step methods. Convergence and error analysis for one-step methods. Practical implementation of one-step methods. Boundary-value problems for ordinary differential equations: shooting methods, finite-difference methods, collocation methods.
A.Quarteroni, R.Sacco, F.Saleri, Matematica Numerica, Springer-Verlag.
G.Naldi, L.Pareschi, G. Russo, Introduzione al Calcolo Scientifico (metodi e applicazioni con Matlab), McGraw-Hill.
G.Monegato, Fondamenti di Calcolo Numerico, CLUT, Torino.
William J. Palm III, Introduction to MATLAB 7 for engineerings, McGraw-Hill.