Krylov Subspace Methods

With respect to the influence on the development and practice of science and engineering in the 20th century, Krylov subspace methods are considered as one of the ten most important classes of numerical methods (see list below). Many scientific problems in three dimensions would be simply intractable without them. Krylov subspace methods are iterative methods that solve large systems of linear equations by avoiding matrix-matrix operations in favour of multiplying vectors by the matrix and then working with the resulting vectors.
Starting from the general idea of iterative solvers for solving partial differential equations I will introduce the concept of Krylov subspace methods, outline the Arnoldi iteration process, which leads to a basis of the Krylov subspace and approximate eigenvalues of large sparse matrices, and discuss one flavour of this class of methods: the generalised minimal residual method (GMRES).
Top 10 algorithms in chronological order according to J. Dongarra, F. Sullivan, The top 10 algorithms, Computing in Science and Engineering, 2, 22-23 (2000).