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Topic: Rayleigh quotient iteration

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	Rayleigh quotient - Wikipedia, the free encyclopedia
		In mathematics, for a given complex Hermitian matrix A and nonzero vector x, the Rayleigh quotient R(A,x) is defined as:
		The Rayleigh quotient is used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation.
		Specifically, this is the basis for Rayleigh quotient iteration.
	en.wikipedia.org /wiki/Rayleigh_quotient (136 words)

	Rayleigh quotient -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		For real matrices and vectors, the condition of being Hermitian reduces to that of being (additional info and facts about symmetric) symmetric, and the (additional info and facts about conjugate transpose) conjugate transpose to the usual (A matrix formed by interchanging the rows and columns of a given matrix) transpose.
		The Rayleigh quotient is used in (additional info and facts about eigenvalue algorithm) eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation.
		Specifically, this is the basis for (additional info and facts about Rayleigh quotient iteration) Rayleigh quotient iteration.
	www.absoluteastronomy.com /encyclopedia/r/ra/rayleigh_quotient.htm (234 words)

	Rayleigh quotient iteration - Wikipedia, the free encyclopedia
		Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates.
		Rayleigh quotient iteration is an iterative method, that is, it must be repeated until it converges to an answer (this is true of all eigenvalue algorithms).
		Fortunately, very rapid convergence is guaranteed; in practice, no more than a few iterations are ever needed.
	en.wikipedia.org /wiki/Rayleigh_quotient_iteration (103 words)

	Eigenvalue algorithm
		In practice, the vector should be normalized after every iteration.
		In addition, some of the better algorithms for the generalized eigenvalue problem are based on power iteration.
		This method can in general be quite slow.
	www.brainyencyclopedia.com /encyclopedia/e/ei/eigenvalue_algorithm.html (378 words)

	Citations: The Rayleigh quotient iteration and some generalizations for nonnormal matrices - Parlett (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		The RQI is of particular interest because of its cubic convergence and its potential use in the shifted QR algorithm [22, 15] In some cases, especially for multiple or clustered eigenvalues, it is advisable to compute the whole invariant subspace spanned by the corresponding eigenvectors.
		The RQI is of particular interest because of its cubic convergence and its potential use in the shifted QR algorithm [Wat82,Par98] In some cases, especially for multiple or clustered eigenvalues, it is advisable to compute the whole invariant subspace spanned by the corresponding eigenvectors.
		In the unsymmetric case we have quadratical convergence [21, 19] In view of the speed of convergence of Shift and Invert methods it may hardly be worthwhile to accelerate them in a Davidson manner: the overhead is significant and the gains may only be minor.
	citeseer.ist.psu.edu /context/265246/0 (3080 words)

	Rayleigh Quotient Iteration
		A natural extension of inverse iteration is to vary the shift at each step.
		In general, Rayleigh quotient iteration (RQI) will need fewer iterations to find an eigenvalue than inverse iteration with a constant shift; it ultimately has cubical convergence, while inverse iteration converges linearly.
		Furthermore, there is the tiny but nasty possibility that it may not converge to an eigenvalue/eigenvector pair at all.
	www.cs.utk.edu /~dongarra/etemplates/node97.html (166 words)

	A Grassmann-Rayleigh Quotient Iteration for Computing Invariant Subspaces - Absil, Mahony, Sepulchre, Van Dooren ... (Site not responding. Last check: 2007-10-08)
		Abstract: The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.
		Rayleigh quotient iteration, invariant subspace, Grassmann manifold AMS subject classification.
		10 Simultaneous Newton's iteration for the eigenproblem (context) - Chatelin - 1984
	citeseer.ist.psu.edu /absil02grassmannrayleigh.html (683 words)

	Well-posedness and regularity properties of the Grassmann-Rayleigh quotient iteration (Site not responding. Last check: 2007-10-08)
		A generalization of the Rayleigh quotient iteration has recently been proposed on the Grassmann manifold.
		This iteration has been shown to converge locally cubically to the invariant subspaces of symmetric matrices.
		Results are obtained e.g.\ concerning fixed points, smoothness, and singularities of the iteration mapping.
	www.montefiore.ulg.ac.be /~absil/Publi/GRQIextended.htm (74 words)

	Convergence Analysis of Inexact Rayleigh Quotient Iteration
		Rayleigh quotient iteration (RQI) is known to converge cubically, and we first analyze how this convergence is affected when the arising linear systems are solved only approximately.
		We introduce a special measure of the relative error made in the solution of these systems and derive a sharp bound on the convergence factor of the eigenpair in a function of this quantity.
		It acts as an inexact RQI method in which the use of iterative solvers is made easier because the arising linear systems involve a projected matrix that is better conditioned than the shifted matrix arising in classical RQI.
	epubs.siam.org /sam-bin/dbq/article/39959 (250 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		It is usually called Rayleigh Quotient iteration: Let x_0 be your guess for an eigenvector (with x_0
		RQI is nothing but a special case of shifted inverse iteration : Take a unit-vector u Repeat Choose a good shift s Replace u by a unit-vector in the direction (A-s)^(-1).u Until...
		The Rayleigh Quotient u^T.A.u is just one possible choice of shift.
	www.math.niu.edu /~rusin/known-math/01_incoming/rayleigh (402 words)

	Matrix eigenvalue problem (Site not responding. Last check: 2007-10-08)
		Except for a set of zero measure, for any initial vector, the result willconverge to an eigenvector corresponding to the dominant eigenvalue.
		Its convergence is slow except for special cases of matrices, and without modification, it canonly find the largest or dominant eigenvalue (and the corresponding eigenvector).
		However, we can understand a few ofthe more advanced eigenvalue algorithms as variations of power iteration.
	www.therfcc.org /matrix-eigenvalue-problem-23769.html (321 words)

	Plan-It Purple Event Details: 'Inverse, Shifted Inverse, and Rayleigh Quotient Iteration as Newton's ... (Site not responding. Last check: 2007-10-08)
		The inverse, shifted inverse, and Rayleigh quotient iterations are well-known algorithms for computing an eigenvector of a symmetric matrix.
		In this talk we demonstrate that each one of these three algorithms can be viewed as a standard form of Newton's method from the nonlinear programming literature.
		Our equivalence result also leads us naturally to an understanding of why the convergence of the Rayleigh quotient iteration is cubic and not just quadratic as expected.
	aquavite.northwestern.edu /cal/pp/eventd.cgi?e=25944 (144 words)

	Inverse iteration (Site not responding. Last check: 2007-10-08)
		In numerical analysis, inverse iteration is an iterative eigenvalue algorithm based on power iteration that achieves superior flexibility and performance.
		The basic idea of power iteration is choosing an initial vector b (either an eigenvector approximation or a random vector) and iteratively calculating
		This is the idea behind Rayleigh quotient iteration.
	www.worldhistory.com /wiki/I/Inverse-iteration.htm (295 words)

	EconPapers: Inexact iterations for the approximation of eigenvalues and eigenvectors
		EconPapers: Inexact iterations for the approximation of eigenvalues and eigenvectors
		Inexact iterations for the approximation of eigenvalues and eigenvectors
		Abstract: The algorithms of inverse iteration and Rayleigh quotient iteration for approximating an eigenpair of a matrix contain a step in which a matrix-vector equation must be solved.
	econpapers.repec.org /paper/dgrkubrem/1996724.htm (158 words)

	cm conference abstract: Brandts (Site not responding. Last check: 2007-10-08)
		This happens for example in the Power Method (on a one-dimensional subspace), in Subspace Iteration and Block Rayleigh Quotient Iteration (on a subspace of a fixed dimension p), the Lanczos and the Arnoldi methods, and also in the Jacobi-Davidson method (on subspaces of variable dimensions).
		In this presentation we will show the consequences of solving the Riccati equation by iterative methods, and using the iterates as new subspace vectors, or as vectors with which to expand a given subspace.
		This will lead to an improvement of Inexact Block Rayleigh Quotient Iteration, and also to new algorithms for invariant subspaces that are block versions (or variations of those) of the Jacobi-Davidson Method.
	www.mgnet.org /mgnet/conferences/CMCIM00/abs/brandts.html (252 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Some relations between (exact and inexact) two-sided Jacobi-Davidson and (exact and inexact) two-sided Rayleigh quotient iteration are given, together with convergence rates.
		Furthermore, we introduce an alternating Jacobi-Davidson process, that can be seen as the Jacobi-Davidson analog of Parlett's alternating Rayleigh quotient iteration.
		Keywords: Jacobi-Davidson, Rayleigh quotient iteration, Ostrowski's two-sided Rayleigh quotient iteration, Parlett's alternating Rayleigh quotient iteration, two-sided Lanczos, correction equation, nonnormal matrix, inexact accelerated Newton method, rate of convergence, generalized eigenproblem, polynomial eigenproblem.
	www.cwru.edu /artsci/math/hochstenbach/abstracts/03hsl_abs.txt (135 words)

	Rayleigh Quotient Iteration. (Site not responding. Last check: 2007-10-08)
		Similar to inverse iteration, the Rayleigh quotient iteration (RQI) method of §4.3 can also be generalized to solve the problem (5.1).
		The only difference between Algorithms 5.2 and 5.3 is in step (8), where the shift is updated.
		The reward for this is a cubic rate of convergence.
	www.cs.utk.edu /~dongarra/etemplates/node168.html (63 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		> # Rayleigh Quotient Iteration w/ deflation > # of complex mxm C, > # to get _ALL_ eigenvalues > # followed by inverse iteration > # to get all eigenvectors.
		> M:=evalf(evalm(C)): > printf(`\n RAYLEIGH QUOTIENT ITERATION w/ DEFLATION\n`); RAYLEIGH QUOTIENT ITERATION w/ DEFLATION > n:=m; n := 5 > for k from 1 to m-1 do > Idn:=diag('1.0' $ ('i'=1..n)); > xv:=randvector(n): > yv:=randvector(n): > zv:=matadd(xv,yv,1,I): > dis:=infinity; > test:=0.; > for it from 1 to 25 > while (dis>1.0 or test
		> X:=evalf(augment(uv,Idn)); > QRdecomp(X,Q='U'); > U:=evalf(evalm(U)); > M:=multiply(htranspose(U),M,U); > M:=submatrix(M,2..n,2..n); > n:=n-1; > od: lam=-43.5766450696602522140604+106.358451291029228444776I, it=12, dis=3.54e-01, test=1.22e+00 lam=-108.067839327715693469085+23.8134396333860157517779I, it=11, dis=2.88e-01, test=2.01e+00 lam=33.8351778994824143244665+108.487127124126530217014I, it=9, dis=4.27e-01, test=3.89e+00 lam=134.899622987860321272466-91.0738602668532689202781I, it=7, dis=2.00e-01, test=1.58e+01 > printf(`\n`); > lambda[m]:=M[1,1]; lambda[5] := 48.9096835100332100862128 - 188.585157781688505493290 I > printf(`\n INVERSE ITERATION.
	www.math.okstate.edu /~burchar/ralqdfl.txt (162 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Note that the arrays holding the eigenvectors and the eigenvalues must be allocated.
		The main difference is that there are multiple states to monitor, that is, many eigenvalue/eigenvector pairs.
		JMP uses eigenvalue transformations to generate methods such as the inverse iteration or Rayleigh quotient iteration.
	www.mi.uib.no /~bjornoh/jmp/0.7.1/jmp/iter/eig/package.html (163 words)

	A Grassmann--Rayleigh Quotient Iteration for Computing Invariant Subspaces
		A Grassmann--Rayleigh Quotient Iteration for Computing Invariant Subspaces: SIAM Review Vol.
		The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A.
		Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.
	epubs.siam.org /sam-bin/dbq/article/37864 (124 words)

	Michigan Applied and Interdisciplinary Mathematics Seminar (Site not responding. Last check: 2007-10-08)
		Interpreting the Classical Inverse, Shifted, Inverse, and Rayleigh Quotient Iteration Methods as a Standard Formulation of Newton's Method from the Nonlinear Programming Literature
		In this talk we demonstrate that each one of these three algorithms can be viewed as a standard form of Newton's method from the nonlinear programming literature, involving an norm projection.
		Our equivalence result also leads us naturally to a new proof that the convergence of the Rayleigh quotient iteration is q-cubic with rate constant at worst 1.
	www.math.lsa.umich.edu /seminars/applied/winter03/mar07.html (119 words)

	Achiya Dax: Computing SVD via the Orthogonal Rayleigh Quotient Iteration ( ORQI ) method (Site not responding. Last check: 2007-10-08)
		Computing SVD via the Orthogonal Rayleigh Quotient Iteration (ORQI) method
		The Orthogonal Rayleigh Quotient Iteration (ORQI) method is an effective tool for calculating complete eigensystems of symmetric tridiagonal matrices.
		In this talk we present a modified ORQI scheme for computing the Singular Value Decomposition (SVD) of a real n x n bidiagonal matrix B.
	osijek.fernuni-hagen.de /~luka/Abstracts/node39.html (165 words)

	Richard A. Tapia - Vitae - Mathematical Presentations
		Invited address, “Inverse, Shifted Inverse, and Rayleigh Quotient Iteration as Newton’s Method”, International Joint Meeting of the AMS and the Sociedad Matemática Mexicana (SMM), Houston, Texas, May 13
		Invited address, “Inverse, Shifted Inverse, and Rayleigh Quotient Iteration as Newton’s Method”, Applied Mathematics and Numerical Analysis Seminar, University of Minnesota, Minneapolis, Minnesota, November 20
		Invited address, “Inverse, Shifted, Inverse, and Rayleigh Quotient Iteration Methods as Newton's Method,” Department of Mathematics, Trinity University, San Antonio, Texas, February 25
	www.caam.rice.edu /~rat/cv/math_presentations.html (1929 words)

	Inexact Iterations (SMEALSearch) - Pal,Rangaswamy,Giles,Debnath (Site not responding. Last check: 2007-10-08)
		The algorithms of inverse iteration and Rayleigh quotient iteration for approximating an eigenpair of a matrix contain a step in which a matrix-vector equation must be solved.
		The behaviour of these algorithms is analysed if this equation is solved only approximately with a known tolerance.
		@misc{ approximation-inexact, author = "For The Approximation", title = "Inexact Iterations" }
	smealsearch2.psu.edu /4577.html (274 words)

	Summary of Activities, FY98
		In a related effort, R. Pozo and B. Miller developed SciMark, a benchmark for numerical computing in Java.
		The benchmark, a Java applet, is a composite of an FFT, a Monte Carlo integration, a sparse matrix multiplication, a Gauss-Seidel iteration and a dense LU decomposition.
		O'Leary and G. Stewart, "New Rayleigh Quotient Method with Applications to Large Eigenproblems," Electronic Transactions on Numerical Analysis.
	math.nist.gov /mcsd/Reports/98/yearly (9943 words)

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