The course will be an introduction to the main methods and techniques for the numerical solution of mathematical problems and to their implementation.
Mathematical Analysis I and II, Geometry and Linear Algebra.
The course will consist of lectures based on traditional teaching including theory and exercises. Some of the lectures will be based on exercise sessions using PC-labs.
The final exam is based on a written test followed by an oral interview. The written test is composed of exercises and theoretical questions. The oral exam is aimed at assessing the knowledge of the theory covered by the course. The oral interview is optional.
Specifically the following topics will addressed:
1) Sources of Errors (model, data, method and machine accuracy), Characteristics of a numerical method (Convergence, Accuracy, Reliability and efficiency), conditioning number;
2) Nonlinear equations: bisection method, Newton method, secant method, fixed point iteration. Stopping and convergence criteria;
3) Linear systems: forward substitution, backward substitution, LU factorization, Gauss elimination method, Cholesky factorization, spectral conditioning number, Richardson method, Jacobi method, Gauss-Seidel method, gradient and conjugate gradient. Stopping and convergence criteria;
4) Approximation of functions: Lagrangian interpolation and error of the Lagrange interpolating polynomial, spline, least squares method;
5) Integration: Riemann sum, trapezoidal rule, Simpson's rule. Composite integration. Newton-Cotes formulas. Gaussian formulas;
6) Resolution of ordinary differential equations: approximation of derivatives through finite differences. Cauchy problem. Explicit, implicit Euler method, Crank-Nicolson method, Heun method. Consistency, stability and convergence analysis. Runge-Kutta and multi-step methods. stiffness;
7) Partial differential equations: Heat equations, finite difference method, problems with boundary values;
8) Numerical computation of eigenvalues: power method, inverse power method, shifted inverse power method.
The course material will be provided by means of the e-learning platform of the University of Bergamo.
If the teaching activity will be mixed or in remote mode, changes can be done compared to what stated in the syllabus, to make the course and the exams available also in these modalities.
For more details write to: francesca.maggioni@unibg.it
