Objectives and Contextualisation

Modeling turns problems in science and engineering into mathematical problems. The complexity of the real world often gives rise to mathematical problems that cannot be addressed from an analytic approach. Or perhaps they can, but the analytic approach may be too complex in the context in which the solution of the problem is required. For instance, solving the problem could be part of a contract, for which limited time is available.

Numerical methods are techniques from which algorithms can be deduced in order to obtain approximate solutions of mathematical problems. Many times, especially when high precision is required, these algorithms demand a large amount of computations. The use of a computer is then mandatory. Computers are most efficient when using finite precision arithmetic (this is, working with a finite number of digits). This means that each operation introduces error, known as round-off. This is not usually a problem, in particular because we are looking for approximate solutions. Nevertheless, it is necessary to know how to avoid situations in which the propagation of round-off error could completely invalidate our computations.

This course is devoted to the analysis of basic numerical methods, related to the solution of the kind of mathematical problems studied in first-year courses. This analysis has as a goal being able to predict both the quality of the approximations produced by the different methods and the computational effort they involve. This course is also an introduction to scientific computing, this is, the set of techniques and skills needed for the implementation in a computer of the numerical solution of a problem.

The numerical methods studied in this course are building stones of numerical methods for the solution of more sophisticated problems, studied from the second year on, like ordinary and partial differential equations.