Lanczos algorithm

Lanczos algorithm

From Wikipedia, the free encyclopedia
The Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to findeigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very large sparse matrices. In Latent Semantic Indexing, for instance, matrices relating millions of documents to hundreds of thousands of terms must be reduced to singular-value form.
Peter Montgomery published in 1995 an algorithm, based on the Lanczos algorithm, for finding elements of the nullspace of a large sparse matrix over GF(2); since the set of people interested in large sparse matrices over finite fields and the set of people interested in large eigenvalue problems scarcely overlap, this is often also called the block Lanczos algorithm without causing unreasonable confusion. See Block Lanczos algorithm for nullspace of a matrix over a finite field.

Lanczos algorithm

Lanczos algorithm

From Wikipedia, the free encyclopedia
The Lanczos algorithm is an iterative algorithm invented by Cornelius Lanczos that is an adaptation of power methods to findeigenvalues and eigenvectors of a square matrix or the singular value decomposition of a rectangular matrix. It is particularly useful for finding decompositions of very large sparse matrices. In Latent Semantic Indexing, for instance, matrices relating millions of documents to hundreds of thousands of terms must be reduced to singular-value form.
Peter Montgomery published in 1995 an algorithm, based on the Lanczos algorithm, for finding elements of the nullspace of a large sparse matrix over GF(2); since the set of people interested in large sparse matrices over finite fields and the set of people interested in large eigenvalue problems scarcely overlap, this is often also called the block Lanczos algorithm without causing unreasonable confusion. See Block Lanczos algorithm for nullspace of a matrix over a finite field.

Arnoldi iteration

Arnoldi iteration

From Wikipedia, the free encyclopedia
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of iterative methods. Arnoldi finds the eigenvalues of general (possibly non-Hermitianmatrices; an analogous method for Hermitian matrices is the Lanczos iteration. The Arnoldi iteration was invented by W. E. Arnoldi in 1951.
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.