Objectives and Contextualisation

In the Numerical Analysis course we will study computational algorithms to solve some of the basic problems that usually appear in the scientific calculation, such as the calculation of the solution of nonlinear equations, the resolution of systems of linear equations and the numerical resolution of differential equations.
The goal of the course is that the student learns about these methods from their mathematical foundation, studying the properties of convergence and stability and estimating errors and  their applicability and possible limitations.
Computer laboratory sessions will have an important role in this course. The sessions are a complement to better understand the numerical methods.  The computer lab sessions allow to show, in different examples, the properties of convergence and stability studied analytically in the seminars. The sessions will also help to compare different numerical methods to solve the same problem. Most examples will be considered at an easy level so they can be done by hand or with a calculator. Keep in mind that real problems are usually much more complicated and cannot be done without using a computer. It is in the later cases where the phenomena described in the theory seminars  are produced.
The procedure we will follow consist of first, understand the methods studied in the seminars, then do a series of exercises by hand or with a calculator in order to master the algorithm and finally, codify the algorithm with which to solve more difficult problems. That is why the theory and problem seminars and the computer laboratory sessions have the same importance.

Capacities or skills to acquire.

• Understand the mathematical foundations of the methods.
• Ability to generate or build the different methods.
• Distinguish the different types of errors introduced by a method and understand how to estimate them.
• Understand convergence criteria for iterative methods.
• Learn how to compare different methods to solve the same problem.
• Ability to choose the most appropriate numerical method (s) to solve a given problem.
• Sufficient skills to implement these methods in the most efficient way.
• Give practical stopping iteration criteria in order to obtain a fixed accuracy.

Develop criteria to detect erroneous results and ability to find the source of errors (ill-conditioned problem, method not suitable for the problem considered, unstable numerical scheme, etc.) and correct them.