Krylov subspace

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In linear algebra the Krylov subspace generated by an n-by-n matrix, A, and an n-vector, b, is the subspace $\mathcal{K}_n$ spanned by the vectors of the Krylov sequence:

$\mathcal{K}_n = \operatorname{span} \, \{ b, Ab, A^2b, \ldots, A^{n-1}b \}. \,$

It is named after Russian applied mathematician and naval engineer Alexei Krylov.

Modern iterative methods for finding one (or a few) eigenvalues of large sparse matrices or solving large systems of linear equations 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.

Because the vectors tend very quickly to become almost linearly dependent, 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 Arnoldi, Lanczos, GMRES (generalized minimum residual) and BiCGSTAB (stabilized biconjugate gradient) methods.