Krylov

In numerical linear algebra, a Krylov subspace is a vector space generated by successively applying a matrix to an initial vector. Krylov subspace methods are a class of iterative methods used to solve large sparse linear systems of equations, eigenvalue problems, and related problems involving matrices.

More formally, given an n x n matrix A and a vector b of length n, the k-th Krylov subspace K_k(A, b) is defined as the span of the vectors:

K_k(A, b) = span{b, Ab, A²b, ..., A^k-1b}

Krylov subspace methods aim to find approximate solutions within this subspace. They are particularly effective when dealing with large sparse matrices, where direct methods become computationally expensive or infeasible. Common Krylov subspace methods include the Conjugate Gradient (CG) method for symmetric positive-definite matrices, the Generalized Minimal Residual method (GMRES) for non-symmetric matrices, and the Lanczos algorithm, which is used for eigenvalue problems.

The efficiency of Krylov subspace methods often depends on the spectral properties of the matrix A. Preconditioning techniques are frequently employed to improve convergence rates by modifying the system to one with more favorable spectral characteristics. The Arnoldi iteration process is often used as a basis for building orthonormal bases for Krylov subspaces, enabling more stable computations.