Editing Arnoldi iteration (section)

{{Short description|Iterative method for approximating eigenvectors}}
In [[Numerical analysis|numerical]] [[linear algebra]], the '''Arnoldi iteration''' is an [[eigenvalue algorithm]] and an important example of an [[iterative method]].  Arnoldi finds an approximation to the [[eigenvalue]]s and [[eigenvector]]s of general (possibly non-[[Hermitian matrix|Hermitian]]) [[Matrix (mathematics)|matrices]] by constructing an orthonormal basis of the [[Krylov subspace]], which makes it particularly useful when dealing with large [[sparse matrix|sparse matrices]].

The Arnoldi method belongs to a class of linear algebra algorithms that give a partial result after a small number of iterations, in contrast to so-called ''direct methods'' which must complete to give any useful results (see for example, [[Householder transformation]]). The partial result in this case being the first few vectors of the basis the algorithm is building.

When applied to Hermitian matrices it reduces to the [[Lanczos algorithm]].  The Arnoldi iteration was invented by [[W. E. Arnoldi]] in 1951.<ref>{{Cite journal |last=Arnoldi |first=W. E. |year=1951 |title=The principle of minimized iterations in the solution of the matrix eigenvalue problem |url=https://www.ams.org/qam/1951-09-01/S0033-569X-1951-42792-9/ |journal=Quarterly of Applied Mathematics |language=en |volume=9 |issue=1 |pages=17–29 |doi=10.1090/qam/42792 |issn=0033-569X|doi-access=free }}</ref>