# Arnoldi iteration

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The term

*iterative method*, used to describe Arnoldi, can perhaps be somewhat confusing. Note that all general eigenvalue algorithms must be iterative. This is not what is referred to when we say Arnoldi is an iterative method. Rather, Arnoldi belongs to a class of linear algebra

algorithms (based on the idea of

Krylov subspaces) that give a partial result after a relatively small number of iterations. This is in contrast to so-called

*direct methods*, which must complete to give any useful results.

Arnoldi iteration is a typical large sparse matrix algorithm: It does not access the elements of the matrix directly, but rather makes the matrix map vectors and makes its conclusions from their images. This is the motivation for building the

Krylov subspace.