This course introduces the basic concepts and methods of Numerical Analysis with the aim of providing reliable and accurate approximations to many problems recurrent in the engineering practice, described via linear and nonlinear mathematical models. The different methods are theoretically derived with particular attention to the key-concepts of convergence, consistency and stability.

The course is organized in lectures and computer labs. Laboratory sessions are meant to provide a practical feedback to the theoretical knowledge with the implementation of the algorithms and methods presented during the lectures. All computations are carried out using the Matlab scientific software environment.

The lectures and computer labs will allow the students, who have successfully passed the exam, to know and understand:­

‐ the basic concepts and the possible errors due to the floating point arithmetics;

- the main methods to approximate linear systems, data and functions, nonlinear equations, derivative and integrals, ordinary differential equations;

- the properties of convergence, consistency and stability for the considered approximation schemes;

- the algorithms coding the schemes introduced to approximate linear systems, data and functions, nonlinear equations, derivative and integrals, ordinary differential equations;

- the use of MATLAB software for the implementation of scripts and functions to code the considered numerical methods.

Thus, the acquired knowledge can be employed for:

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- the implementation of numerical algorithms for solving common problems in engineering practice that require the solution of linear systems, the approximation of data and functions, nonlinear equations, the integration and the differentiation of functions, ordinary differential equations;

- critical reasoning and interpretation of the obtained results;

‐ a shrewd choice of the numerical method best suited to the solution of basic mathematical problems such as the solution of linear systems, the approximation of data and functions, nonlinear equations, the integration and the differentiation of functions, ordinary differential equations.

The student is expected to show a critical understanding of the contents of the course, without limiting to the presentation of definitions and results. Furthermore, students are expected to be able to solve the exercises with a rigorous and logical approach consistently with the theoretical knowledge.

- Floating-point arithmetics: different sources of the computational error; absolute vs relative errors; the floating point representation of real numbers; the roundoff unit; the machine epsilon; floating-point operations; over- and under-flow; numerical cancellation;

- Numerical approximation of nonlinear equations: the bisection and the Newton methods; the fixed-point iteration; convergence analysis (global and local results); order of convergence; stopping criteria and corresponding reliability; generalization to system of nonlinear equations (hints);

- Numerical approximation of systems of linear equations: direct methods (Gaussian elimination method; LU factorization; pivoting; Cholesky factorization; sparse systems: Thomas algorithm for tridiagonal systems); iterative methods (the stationary and the dynamic Richardson scheme; Jacobi, Gauss-Seidel, gradient, conjugate gradient methods; choice of the preconditioner; stopping criteria and corresponding reliability); accuracy and stability of the approximation; the condition number of a matrix; over- and under-determined systems: the singular value decomposition;

- Numerical approximation of functions and data: Lagrange form of the polynomial interpolation; piecewise interpolation; cubic interpolating splines; least-squares approximation of clouds of data;

- Numerical approximation of derivatives: finite difference schemes of first and second order; the undetermined coefficient method;

- Numerical approximation of definite integrals: simple and composite formulas; midpoint, trapezoidal, Cavalieri-Simpson quadrature rules; Gaussian formulas; degree of exactness and order of accuracy of a quadrature rule;

- Numerical approximation of ODEs: the Cauchy problem; one-step methods (forward and backward Euler and Crank-Nicolson schemes); consistency, stability and convergence.

Knowledge of Mathematical Analysis, Calculus and Linear Algebra are required.

Five exam sessions are scheduled, on the dates established by the Faculty calendar.

The exam consists of a written test during which it will be required:

• to answer to theoretical questions on the topics covered during the course, which may deal with the statement and/or the proof of relevant theoretical results; the precise definition of concepts such as convergence, consistency, stability of a numerical scheme; the deduction of a numerical method;
• to use built-in MATLAB functions and MATLAB functions implemented during computer labs for the solution of numerical problems, including systems of linear equations, interpolation of data and functions, nonlinear equations, numerical integration and differentiation, approximation of ordinary differential equations.

The exam evaluation will take into account the correctness and the accuracy of the provided answers, the critical reasoning capacity, the ability to implement MATLAB algorithms and to use the MATLAB functions implemented during computer labs to solve basic mathematical problems as well as problems of interest in engineering practice.