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
[3] Krylov subspaces are used in algorithms for finding approximate solutions to high-dimensional linear algebra problems.
[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.
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
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).