The student will get familiar with the basic algorithms for solving the main four problems of numerical linear algebra (NLA), namely: (1) the solution of linear systems, (2) the solution of least squares problems, (3) the computation of eigenvalues and eigenvectors, and (4) the computation of the singular value decomposition (SVD). Also, he/she will acquire techniques and tools from NLA that can be useful either in his/her professional performance, in areas like data analysis and pattern recognition, or in a scientific career in the field of applied and computational mathematics. In particular, the student will learn and will be able to manage:
- The basic MATLAB commands in the context of the four main problems in NLA mentioned above.
- The fundamentals of numerical analysis (conditioning, stability, and computational complexity).
- The error analysis in numerical methods, in particular those appearing in NLA.
- The basic facts of the floating point arithmetic.
- The basic notions of matrix norms, together with their relevance in the numerical computations that involve the use of matrices.
- The tools and the theory underlying the algorithms that are currently employed for the solution of linear systems, both for matrices with small to moderate size (direct methods) and for large scale matrices (iterative methods).
- The tools and the theory underlying the algorithms that are currently employed for the computation of eigenvalues and eigenvectors, both for matrices with small to moderate size (direct methods) and for large scale matrices (iterative methods).
- The theory and tools in the computation of the SVD, as well as an approach to the basic algorithms for computing such decomposition.
- The basic theory and tools in the solution of least squares problems.
- Some of the applications of the SVD in both theoretical and applied frameworks, like the distance to the set of matrices with smaller rank or the image compression and the principal component analysis.
- Some of the standard applications of NLA in applied contexts, like data analysis or image recognition.
Competences associated with this subject:
In this subject, the student will make progress in achieving the following competences that are indicated in the ¿Memoria de verificación¿ of the Master:
CB6: Having and understanding the knowledge that provides a basis or opportunity to be original in the development and/or application of ideas, often in a research context.
CB7: Students know how to apply their acquired knowledge and problem-solving skills in new or unfamiliar settings within broader (or multidisciplinary) contexts related to their field of study.
CB8: Que los estudiantes sean capaces de integrar conocimientos y enfrentarse a la complejidad de formular juicios a partir de una información que, siendo incompleta o limitada, incluya reflexiones sobre las responsabilidades sociales y éticas vinculadas a la aplicación de sus conocimientos y juicios. Students are able to integrate knowledge and to face the complexity of making judgments based on information that, being incomplete or limited, includes reflections on the social and ethical responsibilities linked to the application of their knowledge and judgments.
CB10: Students have the learning skills that will enable them to continue studying in a way that will be largely self-directed or autonomous.
CG1: Collect and interpret data of a mathematical nature which can be applied to other domains of scientific knowledge.
CG2: Apply acquired knowledge and possess the ability to solve novel problems related with Mathematics.
CG4: Being able to generate new ideas which may imply an advance of knowledge for Mathematics
CG5: Being able to communicate conclusions in clear and precise way.
CG6: Being able to autonomously study and do research.
CG7: Being able to do team-work and manage available time.
CE1: Understanding and properly using mathematical language.
CE2: Being able to formulate mathematical statements in various fields and set up proofs.
CE3: Being able to abstract structural properties differentiating them from more accidental ones.
CE4: Being able to solve mathematical problems, planning their solution in terms of the available tools and of additional time and resource limitations.
CE5: Being able to develop computer software which solves mathematical problems using the most suitable computational environment in each case.
CE6: Being able to design and implement more or less complex algorithms to solve real-life problems.
CE8: Being able to reflect on obtained results, formulating their domain of validity and/or applicability.
CE11: Being able to understand and apply advanced knowledge on numerical methods and computing to problems in science, technology, and society.