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Krylov subspace

( Numerical Linear Algebra )

Background
Properties
Use
Issues
Existing methods
See also
References
Further reading

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Krylov Linear Mathcal Subspace

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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 Linear span by the images of b under the first r powers of A (starting from $A^0=I$ ), that is,

Jorge Nocedal (2025). 9780387303031, Springer. ISBN 9780387303031

\mathcal{K}_r(A,b) = \operatorname{span} \, \{ b, Ab, A^2b, \ldots, A^{r-1}b \}.

Background

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

Properties

$\mathcal{K}_r(A,b), A\,\mathcal{K}_r(A,b)\subset \mathcal{K}_{r+1}(A,b)$ .
Let $r_0 = \operatorname{dim} \operatorname{span} \, \{ b, Ab, A^2b, \ldots \}$ . Then $\{ b, Ab, A^2b, \ldots, A^{r-1}b \}$ are linearly independent unless $r>r_0$ , $\mathcal{K}_r(A,b) \subset \mathcal{K}_{r_0}(A,b)$ for all $r$ , and $\operatorname{dim} \mathcal{K}_{r_0}(A,b) = r_0$ . So $r_0$ is the maximal dimension of the Krylov subspaces $\mathcal{K}_r(A,b)$ .
The maximal dimension satisfies $r_0\leq 1 + \operatorname{rank} A$ and $r_0 \leq n$ .
Consider $\dim \operatorname{span} \, \{ I, A, A^2, \ldots \} = \deg\,p(A)$ , where $p(A)$ is the minimal polynomial of $A$ . We have $r_0\leq \deg\,p(A)$ . Moreover, for any $A$ , there exists a $b$ for which this bound is tight, i.e. $r_0 = \deg\,p(A)$ .
$\mathcal{K}_r(A,b)$ is a cyclic submodule generated by $b$ of the torsion $kx$ -module $(k^n)^A$ , where $k^n$ is the linear space on $k$ .
$k^n$ can be decomposed as the direct sum of Krylov subspaces.

Use

Krylov subspaces are used in algorithms for finding approximate solutions to high-dimensional linear algebra problems. 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 Gram matrix associated with the system/output maps so the uncontrollable and unobservable subspaces are simply the orthogonal complement to the Krylov subspace.

Modern such as Arnoldi iteration can be used for finding one (or a few) eigenvalues of large sparse matrix 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 $b$ , one computes $A b$ , then one multiplies that vector by $A$ to find $A^2 b$ 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 $A$ , giving rise to Matrix-free methods.

Issues

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 matrix or Arnoldi iteration for more general matrices.

Existing 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).

Krylov subspace

( Numerical Linear Algebra )

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Krylov subspace ( Numerical Linear Algebra )

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Krylov subspace

( Numerical Linear Algebra )