Chapter

Krylov Subspace Methods

Jörg Liesen and Zdenek Strakos

in Krylov Subspace Methods

Published in print October 2012 | ISBN: 9780199655410
Published online January 2013 | e-ISBN: 9780191744174 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199655410.003.0002

Series: Numerical Mathematics and Scientific Computation

Krylov Subspace Methods

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This chapter derives, under natural conditions and assumptions, the main ideas of Krylov subspace methods from a general projection framework. Incorporating the Krylov subspaces as search spaces in the projection process ensures, by construction, the mathematical finite termination property. Specific choices of the search and constraint spaces characterise the particular methods. Search and constraint spaces are built up by their bases, typically using either the Arnoldi or the Lanczos algorithm. The chapter discusses mathematical reasons for using orthogonal bases and orthogonal transformations. Some basic Krylov subspace methods are then derived and presented in their algorithmic forms. The conjugate gradient method (CG) is linked with the Galerkin finite element method and it is presented as the variational method minimising the discretised energy related quadratic functional. The end of the chapter stresses that Krylov subspace methods behave highly nonlinearly because they are based on projections onto nonlinear subspaces.

Keywords: projection methods; Petrov–Galerkin framework; orthogonal projections; oblique projections; Arnoldi algorithm; Lanczos algorithm; Galerkin finite element method; CG; SYMMLQ; MINRES; GMRES; A. N. Krylov

Chapter.  31638 words.  Illustrated.

Subjects: Applied Mathematics

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