Learning outcomes of the course unit
Numerical analysis, in essence, is a branch of mathematics which deals with the numerical and therefore constructive solution of problems, formulated and studied in other branches of mathematics. 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. Roundoff Errors and Floating-point Arithmetic.
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 - Euler's method, Runge-Kutta methods - Convergence and stability - Stiff Equations - Boundary problems
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.
Oral lesson and laboratory
Oral exam and practical test