Numerical Linear Algebra (NLA) is a subarea of Applied Mathematics. It is mainly concerned with the development, implementation and analysis of numerical algorithms for solving linear problems. In general, such linear problems arise when discretising a continuous problem by restricting it to a finitedimensional subspace of the original solution space. Hence, the development and analysis of numerical algorithms is almost always problem-dependent. The more is known about the underlying problem, the better a suitable algorithm can be developed.
Nonetheless, many of the so-derived methods are more general in the sense that they can be applied to larger classes of problems than initially intended. One of the challenges in Mathematics is deciding how to describe the necessary assumptions, under which a certain method works, in the most general way. In the context of NLA, this means finding for each method the most general description of matrices to which the method can be applied. It also means extracting the most general methods from the vast number of available algorithms. Particularly for users with new problems this is crucial, as it allows them to apply and test well-established algorithms first, before starting to develop new methods or to extend existing ones.
In this book, I have attempted to use this matrix-driven approach rather than the problem-driven one. Naturally, the selection of the material is biased by my own point of view. Also, a book on NLA without any examples would be rather dire, so there are typical examples and applications included to illustrate the methods, but I have tried to restrict myself to simple examples, which do not require much previous knowledge on specific problems and discretisation techniques.
During the past years, I have given courses on Numerical Linear Algebra at advanced BSc and early MSc level at the University of Sussex (UK), the University of Oxford (UK) and the University of Bayreuth (Germany). I have also given courses on Numerical Analysis which covered parts of the NLA material in Oxford, Göttingen (Germany) and Bayreuth.
This book on Numerical Linear Algebra is based on these courses and the material of these courses. It covers the standard material, as well as more recent and more specific techniques, which are usually not found in standard textbooks on NLA. Examples include the multigrid method, the domain decomposition method, multipole expansions, hierarchical matrices and applications to compressed or compressive sensing.