Linear subspace generated from a vector acted on by a power series of a matrix
In
linear algebra, the order-rKrylov subspace generated by an n-by-nmatrixA and a vector b of dimension n is the
linear subspacespanned by the
images of b under the first r powers of A (starting from ), that is,[1][2]
Background
The concept is named after Russian applied mathematician and naval engineer
Alexei Krylov, who published a paper about the concept in 1931.[3]
Properties
.
Let . Then are linearly dependent unless , for all , and . So is the maximal dimension of the Krylov subspaces .
The maximal dimension satisfies and .
Consider , where is the
minimal polynomial of . We have . Moreover, for any , there exists a for which this bound is tight, i.e. .
is a cyclic submodule generated by of the
torsion-module , where is the linear space on .
Krylov subspaces are used in algorithms for finding approximate solutions to high-dimensional
linear algebra problems.[2] Many
linear dynamical system tests in
control theory, especially those related to
controllability and
observability, involve checking the rank of the Krylov subspace. These tests are equivalent to finding the span of the
Gramians associated with the system/output maps so the uncontrollable and unobservable subspaces are simply the orthogonal complement to the Krylov subspace.[4]
Modern
iterative methods such as
Arnoldi iteration can be used for finding one (or a few) eigenvalues of large
sparse matrices or solving large systems of linear equations. They try to avoid matrix-matrix operations, but rather multiply vectors by the matrix and work with the resulting vectors. Starting with a vector , one computes , then one multiplies that vector by to find and so on. All algorithms that work this way are referred to as Krylov subspace methods; they are among the most successful methods currently available in numerical linear algebra. These methods can be used in situations where there is an algorithm to compute the matrix-vector multiplication without there being an explicit representation of , giving rise to
Matrix-free methods.
The best known Krylov subspace methods are the
Conjugate gradient,
IDR(s) (Induced dimension reduction),
GMRES (generalized minimum residual),
BiCGSTAB (biconjugate gradient stabilized),
QMR (quasi minimal residual),
TFQMR (transpose-free QMR) and
MINRES (minimal residual method).
See also
Iterative method, which has a section on Krylov subspace methods
References
^Nocedal, Jorge; Wright, Stephen J. (2006). Numerical optimization. Springer series in operation research and financial engineering (2nd ed.). New York, NY: Springer. p. 108.
ISBN978-0-387-30303-1.
^
abSimoncini, Valeria (2015), "Krylov Subspaces", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, pp. 113–114
^Hespanha, Joao (2017), Linear Systems Theory, Princeton University Press
Further reading
Nevanlinna, Olavi (1993). Convergence of iterations for linear equations. Lectures in Mathematics ETH Zürich. Basel: Birkhäuser Verlag. pp. viii+177 pp.
ISBN3-7643-2865-7.
MR1217705.
Gerard Meurant and Jurjen Duintjer Tebbens: ”Krylov methods for nonsymmetric linear systems - From theory to computations”, Springer Series in Computational Mathematics, vol.57, (Oct. 2020).
ISBN978-3-030-55250-3, url=
https://doi.org/10.1007/978-3-030-55251-0.
Iman Farahbakhsh: "Krylov Subspace Methods with Application in Incompressible Fluid Flow Solvers", Wiley,
ISBN978-1119618683 (Sep., 2020).