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About 1,284 papers for “Lanczos algorithm”

Nested Lanczos: implicitly restarting a Lanczos algorithm

Gorik De Samblanx, Adhemar Bultheel · Lirias (KU Leuven) · 1997

In this text, we present a generalisation of the idea of the Implicitly Restarted Arnoldi method to the nonsymmetric Lanczos algorithm, using the two-sided Gram-Schmidt process or using a full Lanczos tridiagonalisation. The Implicitly Restarted Lanczos method can be combined with an implicit filter. It can also be used in case of breakdown and offers an alternative for look-ahead.

Cited by 1Open access

Lanczos algorithm explained in statistics

Qiang Niu, M. L. Chen, Joseph Wu · Results in Applied Mathematics · 2025

The Lanczos algorithm is a well-known three-term recurrence that can be used to generate an orthogonal basis for a Krylov subspace derived by a symmetric matrix. In the paper, we present a statistical interpretation of the entries of the tridiagonal matrix generated by the Lanczos process with a diagonal matrix X and an initial vector e . We show that the entries on the main diagonal line can be interpreted as weighted mean and the entries on the super-diagonal line can be understood as weighted sum of variance . Besides, a recurrence for producing the entries on the off-diagonal entries of the tridiagonal matrix is discovered, which leads to a new implementation of the Lanczos process. Finally, numerical examples are provided to investigate the preservation of orthogonality and efficiency in data fitting.

Cited by 0Open access

A Lanczos Algorithm with Restarts

Hermann G. Matthies · Birkhäuser Basel eBooks · 1987

The Lanczos algorithm, originally devised to tridiagonalize a matrix, is used for the generalized eigenvalue problem to operate on a whole subspace, yielding a block-tridiagonal matrix in the Krylov sequence of subspaces. To use prior information and enhance convergence, a subspace restart formulation is introduced. Global orthogonality of the iterated vectors is maintained by a double Gram-Schmidt re-orthogonalization.

Cited by 0

Lanczos Algorithm and Conjugate Gradient

li, hongdali · ResearchWorks at the University of Washington (University of Washington) · 2022

Thesis (Master's)--University of Washington, 2022

Cited by 0Open access

Misconvergence in the Lanczos algorithm

Beresford Ν. Parlett · 1990

Abstract The Lanczos algorithm generates Ritz values in order to approximate eigenvalues. If some eigenvalues are clustered then a Ritz value may hover at a wrong value for a good number of steps. We study this phenomenon and focus on the point of discovery, the first step at which it is certain that there is a hidden eigenvalue in the vicinity of stabilized Ritz values. Both before and after this point the Ritz value behaviour is routine, but for different eigenvalue configurations.

Cited by 6

The Lanczos Algorithm with Implicit Deflation.

Beresford Ν. Parlett, D. Scott · Defense Technical Information Center (DTIC) · 1977

The simple Lanczos process is very effective for finding a few extreme eigenvalues of a large symmetric matrix along with the associated eigenvectors. Unfortunately the process computes redundant copies of the outermost eigenvectors and has to be used with some skill. In this paper it is shown how a modification called implicit deflation stifles the formation of duplicate eigenvectors without increasing the cost of a Lanczos step significantly. The degree of linear independence among the Lanczos vectors is controlled without the costly process of reorthogonalization. (Author)

Cited by 1

The Lanczos Algorithm With Partial Reorthogonalization

Horst D. Simon · Mathematics of Computation · 1984

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly without computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.

Cited by 48

The Lanczos algorithm and Padé approximations

Paul Van Dooren · 1995

Introduction In these two lectures we try to show the relations between the Lanczos algorithm and Pad'e approximations as used e.g. in identification and model reduction of dynamical systems. These notes are based on material in the papers [10, 17, 11, 12] for which a lot of credit ought to be given to the respective coauthors. For simplicity we assume here that all systems are SISO, although some results do extend to the MIMO case. Let a n-th order dynamical system be described by x = Ax + bu (1) y = cx + du (2) where A is a square, b is a column vector, c is a row vector, and d is a scalar. It is well-known that the transfer function of this system : h(s) = c(sI \\Gamma A) \\Gamma1 b +<F29

Cited by 0

A Lanczos bidiagonalization algorithm for Hankel

Kevin Browne A, Sanzheng Qiao A, Yimin Wei B · 2007

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Available online at www.sciencedirect.com

Cited by 0

The Lanczos algorithm with partial reorthogonalization

Horst D. Simon · Mathematics of Computation · 1984

The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly <italic>without</italic> computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.

Cited by 279Open access

On Lanczos’ Algorithm for Tridiagonalizing Matrices

Robert L. Causey, Robert T. Gregory · SIAM Review · 1961

Previous article Next article On Lanczos’ Algorithm for Tridiagonalizing MatricesR. L. Causey and R. T. GregoryR. L. Causey and R. T. Gregoryhttps://doi.org/10.1137/1003063PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards, 45 (1950), 255–282 MR0042791 CrossrefISIGoogle Scholar[2] Robert T. Gregory, Results using Lanczos' method for finding eigenvalues of arbitrary matrices, J. Soc. Indust. Appl. Math., 6 (1958), 182–188 10.1137/0106011 MR0103584 0085.10805 LinkISIGoogle Scholar[3] J. H. Wilkinson, The calculation of eigenvectors by the method of Lanczos, Comput. J, 1 (1958), 148–152 10.1093/comjnl/1.3.148 MR0102915 0117.11002 CrossrefISIGoogle Scholar[4] Heinz Rutishauser, Beiträge zur Kenntnis des Biorthogonalisierungs-Algorithmus von Lanczos, Z. Angew. Math. Physik, 4 (1953), 35–56 10.1007/BF02075304 MR0055307 0050.01008 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Lanczos Methods for the Solution of Nonsymmetric Systems of Linear EquationsWayne JoubertSIAM Journal on Matrix Analysis and Applications, Vol. 13, No. 3 | 17 July 2006AbstractPDF (2194 KB)Some History of the Conjugate Gradient and Lanczos Algorithms: 1948–1976Gene H. Golub and Dianne P. O’LearySIAM Review, Vol. 31, No. 1 | 2 August 2006AbstractPDF (6558 KB)On the similarity transformation to tridiagonal formLinear and Multilinear Algebra, Vol. 12, No. 1 | 30 May 2007 Cross Ref On Lanczos' algorithm for tri-diagonalizationHiroshima Mathematical Journal, Vol. 32, No. 2 | 1 Jan 1968 Cross Ref Volume 3, Issue 4| 1961SIAM Review293-368 History Submitted:20 July 1961Published online:18 July 2006 InformationCopyright © 1961 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1003063Article page range:pp. 322-328ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

Cited by 9

Lanczos algorithm for symmetric eigenvalue problems

David Scott · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 2014

The Lanczos algorithm was originally used to tridiagonalize symmetric matrices, but it was soon replaced by more effective methods based on explicit orthogonal similarity transformations. The algorithm was later revived as an effective scheme for solving sparse symmetric eigenvalue problems. The historical development and the current state of the Lanczos algorithm are surveyed.

Cited by 0Open access

Use of Lanczos Algorithm in Elastic Stability

Nicholas F. Morris · Journal of Structural Engineering · 1990

The Lanczos algorithm is applied to the eigenvalue problems of elastic stability, wherein the geometric stiffness matrix is used to describe the effect of the stress resultants. This matrix need not be positive or positive definite for a general structural system. Despite this fact, it is found that a simple algorithm, without the use of sophisticated convergence criteria and its concomitant programming, can be employed in stability computations. Even though the procedure exhibits an unusual convergence pattern, it can solve for the important buckling loads without the use of any a priori assumptions concerning the eigenvalues or eigenvectors. The simple algorithm can also be applied to dynamic eigenvalue problems but, as the example considered herein demonstrates, it may not be efficient in computing all required eigenvalues for a given frequency range.

Cited by 3