Krylov subspace

Linear subspace generated from a vector acted on by a power series of a matrix
In linear algebra, the order r Krylov subspace generated by an n by n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting from ), that is,

The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about the concept in 1931.

. Let Then are linearly independent 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 can be decomposed as the direct sum of Krylov subspaces.

Krylov subspaces are used in algorithms for finding approximate solutions to high dimensional linear algebra problems. 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.

Because the vectors usually soon become almost linearly dependent due to the properties of power iteration, methods relying on Krylov subspace frequently involve some orthogonalization scheme, such as Lanczos iteration for Hermitian matrices or Arnoldi iteration for more general matrices.

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).