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The power and Arnoldi methods in an algebra of circulants [article]

David F. Gleich, Chen Greif, James M. Varah
2011 arXiv   pre-print
This interpretation provides many generalizations of results from matrix or vector-space algebra. We derive the power and Arnoldi methods in this algebra.  ...  These extensions are straightforward in an algebraic sense, but the implications are dramatically different from the standard matrix case.  ...  An interesting topic for future work is exploring other rings besides the ring of circulants. One obvious candidate is the ring of symmetric circulant matrices.  ... 
arXiv:1101.2173v1 fatcat:5cjywtapybhp3azoxru6dksrmy

The power and Arnoldi methods in an algebra of circulants

David F. Gleich, Chen Greif, James M. Varah
2012 Numerical Linear Algebra with Applications  
We derive the power and Arnoldi methods in this algebra. In the course of our derivation, we define inner products, norms, and other notions.  ...  These extensions are straightforward in an algebraic sense, but the implications are dramatically different from the standard matrix case.  ...  The power method Please see the left half of Figure 3 for the sequence of operations in the power method in the circulant algebra.  ... 
doi:10.1002/nla.1845 fatcat:ajzmkbp5jfcozn6mpnhrl3gmsi

GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems

Anne Greenbaum, Lloyd N. Trefethen
1994 SIAM Journal on Scientific Computing  
The GMRES and Arnoldi algorithms, which reduce to the CR and Lanczos algorithms in the symmetric case, both minimize p(A)b over polynomials p of degree n.  ...  Investigation of these true and ideal approximation problems gives insight into how fast GMRES converges and how the Arnoldi iteration locates eigenvalues.  ...  We are happy to acknowledge discussions with Michael Overton, who showed us how the ideal Arnoldi and GMRES problems relate to more general problems of minimization of singular values of functions of matrices  ... 
doi:10.1137/0915025 fatcat:iqm5cthqsjdspervtu4atasuoi

Page 2825 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews  
Algebraic theory and a power method. (English summary) SIAM J. Sci. Comput. 14 (1993), no. 5, 1089-1106. This paper first shows the applications of multivariate eigenvalue problems in statistics.  ...  The convergence of the gener- alized power method is established under general conditions, and related future research topics are outlined. F.  ... 

Page 2880 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
Summary: “In this paper, by introducing a new definition of vector product and quotient and using the ODE numerical solution method, we present a new method for solving positive real systems of linear  ...  As for the Arnoldi method, RKS can be restarted implicitly, using the QR decomposition of a Hessenberg matrix. This restart comes with a projection of the subspace using a rational function.  ... 

The tensor t-function: a definition for functions of third-order tensors [article]

Kathryn Lund
2019 arXiv   pre-print
A complexity analysis for these methods in the context of tensors is also provided.  ...  To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third-order tensors and computed for a small-scale example via block Krylov subspace methods  ...  The author would like to thank Misha Kilmer for useful conversations and the images used in Figure 2 .1, Andreas Frommer and Daniel B.  ... 
arXiv:1806.07261v2 fatcat:6fbqtctvmvd45mnyprlskwpzfe

Projection assisted Dynamic Mode Decomposition of large scale data [article]

Mohammad N. Murshed, M. Monir Uddin
2020 arXiv   pre-print
Here, we design an optimized version of DMD that utilizes time delay coordinates and a projection matrix.  ...  We have deluge of data in time series format for numerous phenomena. The number of snapshots, resolution and many other factors come into play as we look to identify the dynamics in a given problem.  ...  Acknowledgement This project is partially funded by Office of Research, North South University. (Grant number: CTRG-19/SEPS/06)  ... 
arXiv:2001.03332v1 fatcat:op4g25ooq5gvlcnkss3jw7pocy

Page 5174 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
Actually, our examples are such that two relaxations on the fine level and one solution of the appropriate aggregated linear system are sufficient in order to get an exact stationary probability vector  ...  The methods differ in flop count, work space, and inherent parallelism. They have in common that they are immune to overflows and use no matrix inversions.  ... 

Effective Approximation for the Semiclassical Schrödinger Equation

Philipp Bader, Arieh Iserles, Karolina Kropielnicka, Pranav Singh
2014 Foundations of Computational Mathematics  
In this paper we sketch an alternative strategy.  ...  This leads to methods that attain high spatial and temporal accuracy and whose cost scales like O(N ) or O(N log N ), where N is the number of degrees of freedom.  ...  Therefore, the introduction of odd powers of ∂ x is fraught with loss of unitarity and stability. An extra ingredient is required in our algorithm!  ... 
doi:10.1007/s10208-013-9182-8 fatcat:c2jb5soycbhybbd5r44cinyzka

Google matrix of Twitter

K. M. Frahm, D. L. Shepelyansky
2012 European Physical Journal B : Condensed Matter Physics  
We argue that this small fraction of the whole number of users, which can be viewed as the social network elite, plays the dominant role in the process of opinion formation on the network.  ...  We construct the Google matrix of the entire Twitter network, dated by July 2009, and analyze its spectrum and eigenstate properties including the PageRank and CheiRank vectors and 2DRanking of all nodes  ...  Such an analysis will allow to understand in an better way the process of information flow and decision making on social networks.  ... 
doi:10.1140/epjb/e2012-30599-6 fatcat:us6widfabffgxkluagezkbm6nm

Computational methods for large-scale inverse problems: a survey on hybrid projection methods [article]

Julianne Chung, Silvia Gazzola
2021 arXiv   pre-print
Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent regularizing properties  ...  In this paper, we provide a practical and accessible introduction to hybrid projection methods in the context of solving large (linear) inverse problems.  ...  Therefore the vectors defining the Krylov solution space are iteration vectors of the power method for computing the largest eigenpair of a matrix, and hence they become increasingly richer in the direction  ... 
arXiv:2105.07221v2 fatcat:vviylanambfmjheocl7e5k3bnq


1999 Linear Algebra and its Applications  
Arnoldi method and an implicitly restarted re®ned Arnoldi algorithm 287 (1999) 191 Johnson, C.R., see Fallat, S.M. 288 (1999) 149 Johnson, C.R., see Glunt, W. 288 (1999) 1 Johnson, C.R. and R.L.  ...  Ullah, Improved biased estimation in an ANOVA model 289 (1999) 3 Alaimia, M.R., Automorphisms of some Banach algebras of analytic functions 298 (1999) 87 Allen, B.M. and J.  ... 
doi:10.1016/s0024-3795(99)00242-6 fatcat:om6rs7swrreoxcawrw4lgzniri

Computing the Exponential of Large Block-Triangular Block-Toeplitz Matrices Encountered in Fluid Queues [article]

D.A. Bini, S. Dendievel, G. Latouche, B. Meini
2015 arXiv   pre-print
To this end, we propose some algorithms which exploit the Toeplitz structure and the properties of generators.  ...  Such algorithms allow to compute the exponential of very large matrices, which would otherwise be untreatable with standard methods.  ...  The method based on embedding and the one based on power series perform an accurate computation but are slightly more expensive.  ... 
arXiv:1502.07533v1 fatcat:vy5brh2c2ndbrdpdiytbiksqw4

Spectral Sets [article]

Catalin Badea, Bernhard Beckermann
2013 arXiv   pre-print
Estimating the norm of functions of matrices is an essential task in numerous fields of pure and applied mathematics, such as (numerical) linear algebra, functional analysis, and numerical analysis.  ...  This is a survey about spectral sets, to appear in the second edition of Handbook of Linear Algebra (L. Hogben, ed.).  ...  Polynomial Arnoldi method. The Arnoldi process is a popular method for approaching f (A)b for some fixed vector b = 0 and large sparse A [Hig08, Section 13.2].  ... 
arXiv:1302.0546v1 fatcat:4jobcjrb7zch7etmzvfsgqz4py

A generalized isometric Arnoldi algorithm

Michael Stewart
2007 Linear Algebra and its Applications  
This paper describes a generalization of the isometric Arnoldi algorithm and shows that it can be interpreted as a structured form of modified Gram-Schmidt.  ...  The dimension of the subspace is analogous to displacement rank in the generalized Schur algorithm. The algorithm is described in terms of projections and inner products.  ...  Acknowledgment The author would like to thank two anonymous referees for their careful reading and for suggestions that substantially improved this paper.  ... 
doi:10.1016/j.laa.2006.12.021 fatcat:y6oqlplujvddzohdqrznhd4bhy
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