About: Lanczos algorithm

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The Lanczos algorithm is a iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an Hermitian matrix, where is often but not necessarily much smaller than . Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability.

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rdfs:label	Lanczos algorithm (en)
rdfs:comment	The Lanczos algorithm is a iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an Hermitian matrix, where is often but not necessarily much smaller than . Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. (en)
sameAs	Lanczos algorithm Lanczos algorithm
dbp:wikiPageUsesTemplate	dbt:Citation_needed dbt:Cite_book dbt:Cite_journal dbt:For dbt:Main dbt:Notetag dbt:R dbt:Reflist dbt:Numerical_linear_algebra
Subject	Numerical linear algebra
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Lanczos_algorithm?oldid=1074039633&ns=0
Wikipage page ID	2593852 (xsd:integer)
page length (characters) of wiki page	43460 (xsd:nonNegativeInteger)
Wikipage revision ID	1074039633 (xsd:integer)
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has abstract	The Lanczos algorithm is a iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an Hermitian matrix, where is often but not necessarily much smaller than . Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. In 1970, Ojalvo and Newman showed how to make the method numerically stable and applied it to the solution of very large engineering structures subjected to dynamic loading. This was achieved using a method for purifying the Lanczos vectors (i.e. by repeatedly reorthogonalizing each newly generated vector with all previously generated ones) to any degree of accuracy, which when not performed, produced a series of vectors that were highly contaminated by those associated with the lowest natural frequencies. In their original work, these authors also suggested how to select a starting vector (i.e. use a random-number generator to select each element of the starting vector) and suggested an empirically determined method for determining , the reduced number of vectors (i.e. it should be selected to be approximately 1.5 times the number of accurate eigenvalues desired). Soon thereafter their work was followed by Paige, who also provided an error analysis. In 1988, Ojalvo produced a more detailed history of this algorithm and an efficient eigenvalue error test. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Lanczos_algorithm
is Wikipage redirect of	Block Lanczos Lanczos iteration Lanczos method
is Link from a Wikipage to another Wikipage of	Numerical linear algebra ARPACK Arnoldi iteration Beresford Parlett Bidiagonalization Generalized minimal residual method Krylov subspace Density matrix renormalization group Hubbard model Hungarian Americans Lippmann–Schwinger equation Lis (linear algebra library) List of numerical analysis topics Conjugate gradient method Continued fraction Cornelius Lanczos Eigenvalue algorithm Eigenvalues and eigenvectors Henk van der Vorst List of algorithms Energy minimization Galahad library Derivation of the conjugate gradient method Exact diagonalization

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