Numerical methods

This course is intended as an introduction to techniques for carrying out numerical computation on computers.
The course serves three main goals:

•  the understanding of basic numerical techniques with the underlying mathematical notions,
• the hability to interpret the reliability of numerical results,
• the programming skills to implement simple numerical algorithms with Python.

This course presents a broad overview of numerical methods, using calculus, algebra and computing science. The student must become aware of the relevant issues in selecting appropriate method and software and using them wisely, in terms of computational cost, numerical accuracy, complexity and stability. To make the presentation concrete and appealing, the programming environment PYTHON is adopted as a faithful companion.
Topics include:

• Error analysis: modelling error, truncation error, convergence and approximation order, floating point number representation (IEEE754).
• Approximation and interpolation: Lagrange polynomials, spline functions, NURBS, orthogonal polynomials, error estimators.
• Numerical integration and differentiation: backward and centered finite difference, midpoint, trapezoidal and Simpson formula, adaptive techniques.
• Ordinary Differential Equations (ODE): Taylor and Runge Kutta methods, predictor-corrector methods, stability on unbounded intervals and perturbation analysis.
• Linear equations: factorization methods and iterative techniques, complexity, computation of eigenvalues.
• Nonlinear equations: bisection and Newton methods, optimisation applications.
• Partial Differential Equations (PDE): boundary value problems (Laplace, heat equation, waves equation), approximation by finite differences.

Lectures in auditorium, supervised exercise and problem sessions, and unsupervised assignments.
Real-life examples using numerical methods
Use of Python software