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

2011
*
arXiv
*
pre-print

This interpretation provides many generalizations

arXiv:1101.2173v1
fatcat:5cjywtapybhp3azoxru6dksrmy
*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. ...##
###
The power and Arnoldi methods in an algebra of circulants

2012
*
Numerical Linear Algebra with Applications
*

We derive

doi:10.1002/nla.1845
fatcat:ajzmkbp5jfcozn6mpnhrl3gmsi
*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*. ...##
###
GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems

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 ...

##
###
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. ...

##
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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]

2019
*
arXiv
*
pre-print

A complexity analysis for these

arXiv:1806.07261v2
fatcat:6fbqtctvmvd45mnyprlskwpzfe
*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. ...##
###
Projection assisted Dynamic Mode Decomposition of large scale data
[article]

2020
*
arXiv
*
pre-print

Here, we design

arXiv:2001.03332v1
fatcat:op4g25ooq5gvlcnkss3jw7pocy
*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) ...##
###
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

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! ...

##
###
Google matrix of Twitter

2012
*
European Physical Journal B : Condensed Matter Physics
*

We argue that this small fraction

doi:10.1140/epjb/e2012-30599-6
fatcat:us6widfabffgxkluagezkbm6nm
*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. ...##
###
Computational methods for large-scale inverse problems: a survey on hybrid projection methods
[article]

2021
*
arXiv
*
pre-print

Iterative

arXiv:2105.07221v2
fatcat:vviylanambfmjheocl7e5k3bnq
*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 ...##
###
Index

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. ...

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

2015
*
arXiv
*
pre-print

To this end, we propose some algorithms which exploit

arXiv:1502.07533v1
fatcat:vy5brh2c2ndbrdpdiytbiksqw4
*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. ...##
###
Spectral Sets
[article]

2013
*
arXiv
*
pre-print

Estimating

arXiv:1302.0546v1
fatcat:4jobcjrb7zch7etmzvfsgqz4py
*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]. ...##
###
A generalized isometric Arnoldi algorithm

2007
*
Linear Algebra and its Applications
*

This paper describes a generalization

doi:10.1016/j.laa.2006.12.021
fatcat:y6oqlplujvddzohdqrznhd4bhy
*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. ...
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