Krylovmetoder

Krylov methods are a class of algorithms used in numerical linear algebra to solve large, sparse systems of linear equations of the form Ax = b, where A is a matrix, x is the unknown vector, and b is a known vector. These methods are particularly useful when the matrix A is very large and cannot be stored or manipulated directly in memory. Instead of working with the full matrix A, Krylov methods work with the matrix-vector product Av, where v is a vector.

The core idea behind Krylov methods is to generate a sequence of vectors, called a Krylov subspace, which are linear combinations of powers of the matrix A applied to an initial vector, typically the right-hand side vector b. The subspace is spanned by the vectors {b, Ab, A^2b, ..., A^(k-1)b} for some integer k. Krylov methods then seek an approximate solution x_k within this subspace that minimizes the residual error ||b - Ax_k||.

Common Krylov methods include the conjugate gradient method (CG) for symmetric positive-definite matrices, the generalized minimal residual method (GMRES) for general matrices, and the transpose-free quasi-minimal residual method (TFQMR) and Bi-conjugate Gradient Stabilized method (BiCGSTAB) for non-symmetric matrices. These methods differ in how they construct and minimize the residual within the generated Krylov subspace. Krylov methods are iterative, meaning they start with an initial guess and refine the solution over a series of steps. Their efficiency depends on how quickly the Krylov subspace captures the essential information needed to approximate the solution, which is often related to the spectral properties of the matrix A.