Numerical Analysis

Synthetic program:

General introduction to the basic concepts of numerical mathematics. Floating point representation of real numbers: examples and problems. Polynomial interpolation: existence and uniqueness, approximation error. Hermite interpolation. Piecewise polynomial interpolation. Approximation of derivatives with finite differences. Polynomial quadrature formulas: midpoint, trapezoidal rule, Simpson. Use of FFT to compute Fourier coefficients. Review of basic concepts of ordinary differential equations theory. One step numerical methods: concepts of consistence, convergence, stability. Runge-Kutta methods (10.8). Examples of multistep methods: Adams-Bashforth, Adams-Moulton, BDF. Numerical methods for the solution of nonlinear equations: bisection, Newton's method. Fixed point method. Numerical methods for linear systems: triangular systems, LU decomposition and Gauss elimination, pivoting. Stability analysis for the numerical solution of linear systems. Introduction to methods for undetermined systems. Numerical methods for nonlinear systems of equations.