Linear subspace generated from a vector acted on by a power series of a matrix
In linear algebra, the order-rKrylov 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 ), that is,[1][2]
^Nocedal, Jorge; Wright, Stephen J. (2006). Numerical optimization. Springer series in operation research and financial engineering (2nd ed.). New York, NY: Springer. p. 108. ISBN 978-0-387-30303-1.
^Simoncini, Valeria (2015), "Krylov Subspaces", in Nicholas J. Higham; et al. (eds.), The Princeton Companion to Applied Mathematics, Princeton University Press, pp. 113–114
linear algebra, the order-r Krylovsubspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of...
methods. Krylovsubspace methods work by forming a basis of the sequence of successive matrix powers times the initial residual (the Krylov sequence)...
non-Hermitian) matrices by constructing an orthonormal basis of the Krylovsubspace, which makes it particularly useful when dealing with large sparse...
equations. The method approximates the solution by the vector in a Krylovsubspace with minimal residual. The Arnoldi iteration is used to find this vector...
_{1}} is merely the maximum on an m {\displaystyle m} -dimensional Krylovsubspace, we trivially get λ 1 ⩾ θ 1 {\displaystyle \lambda _{1}\geqslant \theta...
\mathbf {p} _{i}} and r i {\displaystyle \mathbf {r} _{i}} span the same Krylovsubspace. Where r i {\displaystyle \mathbf {r} _{i}} form the orthogonal basis...
Lanczos algorithm, although both approximations will belong to the same Krylovsubspace. Extreme simplicity and high efficiency of the single-vector version...
large. Different approaches exist to stabilise Parareal, one being Krylov-subspace enhanced Parareal. There are multiple algorithms that are directly...
method, the boundary element method for solving integral equations, Krylovsubspace methods. We first introduce and illustrate the Galerkin method as being...
The Minimal Residual Method or MINRES is a Krylovsubspace method for the iterative solution of symmetric linear equation systems. It was proposed by...
until the mid-1990s, when Krylovsubspace methods were applied to the problem. The application of preconditioned Krylovsubspace methods allowed much larger...
_{(n-1)i}x^{n-1}.} Frobenius endomorphism Cayley–Hamilton theorem Krylovsubspace Horn, Roger A.; Charles R. Johnson (1985). Matrix Analysis. Cambridge...
variants such as the conjugate gradient squared method (CGS). It is a Krylovsubspace method. Unlike the original BiCG method, it doesn't require multiplication...
algorithms look at the whole subspace generated by the vectors b k {\displaystyle b_{k}} . This subspace is known as the Krylovsubspace. It can be computed by...
either stationary methods such as successive overrelaxation or Krylovsubspace methods. Krylov methods such as GMRES, typically used with preconditioning...
choice of smoothing operators are extremely diverse as they include Krylovsubspace methods and can be preconditioned. Any geometric multigrid cycle iteration...
it pioneered the use of Krylovsubspace methods. The use of shooting methods gave SpectreRF remarkable robustness and the Krylov methods gave it capacity...
2021-08-09. Watkins, David S. (2007). The Matrix Eigenvalue Problem: GR and KrylovSubspace Methods. Philadelphia, PA: SIAM. ISBN 978-0-89871-641-2. Parlett, Beresford...
for their paper Efficient steady-state analysis based on matrix-free Krylov-subspace methods. Verilog-A Language Reference Manual Verilog-AMS Language Reference...
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