Aims and learning outcomes :

This course introduces the basic methods of Numerical Analysis with the aim of providing reliable and accurate approximations to many problems in the engineering practice, described via linear and nonlinear mathematical models. The different methods are theoretically derived with particular attention to the three key-concepts of convergence, consistency and stability. Matlab laboratory sessions at the computer are also included in this course, with the aim of providing the students with a numerical counterpart of the theoretical approaches and with a practical feedback on applications.

Topics :

- The different sources of the computational error: the floating point arithmetic.

- Nonlinear equations: bisection; the Newton method; the fixed point iteration; order of convergence; stopping criteria.

- Linear systems. Direct methods: Gaussian elimination; LU factorization; pivoting; condition number; stability analysis. Sparse matrices: tridiagonal systems. Iterative methods: the Richardson scheme; Jacobi and Gauss-Seidel methods; stopping criteria. SVD decomposition.

- Approximation of data and functions: interpolation versus least-square approximation; piecewise approximation; spline functions; trigonometric approximation.

- Quadrature formulas: the rectangle, trapezoidal and Simpson rules; order of convergence and degree of accuracy.

- Numerical approximation of ordinary differential equations: the Cauchy problem. One step methods: forward and backward Euler methods; the Crank-Nicolson scheme; consistency, stability and convergence. High order methods: Runge-Kutta schemes.

Labs : 

Most of the methods presented during the lectures will be numerically investigated during the laboratory sessions, via Matlab/Octave programming.

Demanded knowledge :

Undergraduate courses on Linear Algebra, Calculus, Mathematical Analysis.

The exam consists of a written test in laboratory, including theory and MATLAB programming, and of a final colloquium. The written test may be substituted by a home project.