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In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method.Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.
The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding Krylov subspace method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term recurrence relation. It can be shown that there is no Krylov subspace method for general ...
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method [1] for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems.
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. They try to avoid matrix-matrix operations, but rather multiply vectors by the matrix and work with the resulting vectors.
Arnoldi iteration — based on Krylov subspaces; Lanczos algorithm — Arnoldi, specialized for positive-definite matrices Block Lanczos algorithm — for when matrix is over a finite field; QR algorithm; Jacobi eigenvalue algorithm — select a small submatrix which can be diagonalized exactly, and repeat
Also, the power method is the starting point for many more sophisticated algorithms. For instance, by keeping not just the last vector in the sequence, but instead looking at the span of all the vectors in the sequence, one can get a better (faster converging) approximation for the eigenvector, and this idea is the basis of Arnoldi iteration. [11]
Walter Edwin Arnoldi (December 14, 1917 – October 5, 1995) was an American engineer mainly known for the Arnoldi iteration, an eigenvalue algorithm used in numerical linear algebra. His main research interests included modelling vibrations, acoustics , aerodynamics of aircraft propeller , and oxygen reclamation problems of space science.
Basic solvers such as residual inverse iteration and successive linear problems. A solver based on polynomial interpolation that relies on PEP solvers. A solver based on rational interpolation (NLEIGS). MFN can be used to compute the action of a matrix function on a vector. A restarted Krylov solver.