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Topic: Rayleigh quotient

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	PlanetMath: Rayleigh quotient
		Namely, one first minimizes the Rayleigh quotient over the whole vector space.
		At each step, one minimizes the Rayleigh quotient over the subspace orthogonal to all the vectors found in the preceding steps to find another eigenvalue and its corresponding eigenvector.
		This is version 6 of Rayleigh quotient, born on 2003-05-27, modified 2006-11-07.
	planetmath.org /encyclopedia/RayleighQuotient.html (301 words)

	John Strutt, 3rd Baron Rayleigh Summary
		Although Rayleigh was content to work at his home, he accepted the position of Cavendish Professor at Cambridge University, due to the fact that revenue from his estate had declined as the result of a general agricultural depression.
		Rayleigh was fellow, secretary and president of the Royal Society of London and received numerous honorary degrees from universities and science academies around the world.
		Rayleigh's continues to be read in the original and cited, whereas the work of most of his contemporaries is not.
	www.bookrags.com /John_William_Strutt%2C_3rd_Baron_Rayleigh (3706 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		This constructs a quotient space of the original seminormed vector space, and this quotient is a normed vector space.
		The quotient of L(H) by K(H) is the Calkin algebra.
		In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.
	www.worldhistory.com /search.php?ix=wikii2&rf=0,PN2,P3,PN4,P6&page=1&type=article&hpp=&lfs=1&q=Quotient (641 words)

	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 (250 words)

	Variation Principle
		Yet, we will see that the Rayleigh quotient provides the means to approximately solve the problem.
		The numerator of the Rayleigh quotient requires the M by M "hamiltonian matrix" defined by
		The objective is to minimize the Rayleigh quotient with respect to
	www.sci.wsu.edu /idea/quantum/variation_principle.htm (642 words)

	Rayleigh quotient iteration
		Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using thee 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 and no more than a few iterations are needed in practice.
	www.algebra.com /algebra/about/history/Rayleigh-quotient-iteration.wikipedia (95 words)

	News \| Gainesville.com \| The Gainesville Sun \| Gainesville, Fla. (Site not responding. Last check: 2007-10-16)
		John William Strutt, 3rd Baron Rayleigh (12 November 1842 – 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904.
		In 1873 his father, the 2nd Baron Rayleigh, died, and he inherited the Barony of Rayleigh.
		Lord Rayleigh was elected to Fellow of the Royal Society on June 12, 1873 and was elected president of the Royal Society between 1905 and 1908.
	www.gainesville.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Lord_Rayleigh (383 words)

	csci 635 lab 14
		The use of the Hessenberg from of a matrix proves to be a very useful way to precondition a matrix before implementing other transformations on it such as using the Rayleigh quotient calculate an approximate eigen value.
		The Rayleigh quotient iteration is a method that combines the calculation of the Raleigh quotient and the inverse quotient at the same time.
		One of the conditions for the algorithm is that we generate a vector v such that the norm of v is 1.
	fadishihadeh.com /class/610/lab14.htm (475 words)

	[No title] (Site not responding. Last check: 2007-10-16)
		From: sci.math.num-analysis@elsner.org (Ulrich Elsner) Subject: Re: Inverse Rayleigh iteration Date: 29 Jan 2001 19:28:13 GMT Newsgroups: sci.math.num-analysis Summary: Rayleigh Quotient iteration to compute eigenvalues Hi, According to Stefan Maehlmann
		It is usually called Rayleigh Quotient iteration: Let x_0 be your guess for an eigenvector (with x_0
		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)

	ME 511, VIBRATIONAL ANALYSIS II
		Rayleigh (proportional damping) leads to uncoupled equations of motion in modal coordinates.
		Rayleigh's quotient constructed from quadratic forms formed from eigenvalue problem for MDOF system.
		Apply Rayleigh's quotient to generate element stiffness and (consistent) mass matrices, and system stiffness and mass matrices.
	www.calstatela.edu /faculty/sfelsze/me511.htm (1399 words)

	Paper ID sheet - Systems and Control Engineering at ULg (Site not responding. Last check: 2007-10-16)
		The continuous-time Rayleigh quotient flow on the sphere
		This interpretation leads to a precise phase portrait for Rayleigh quotient flow.
		In particular, it is shown that complete solutions of the Rayleigh quotient flow visits the eigenvectors of $A$ in ascending order.
	www.montefiore.ulg.ac.be /systems/Publi/rqf.htm (129 words)

	PlanetMath: Rayleigh-Ritz theorem
		Then its eigenvectors are the critical points (vectors) of the "Rayleigh quotient", which is the real function
		for the Rayleigh quotient satisfies the complex eigenvalue equation
		Cross-references: bounds, eigenvalues of a Hermitian matrix are real, unitary matrix, diagonalizable, normal, canonical, relations, complex, satisfies, stationary, column vector, transposition, similar, Calculus, matrix, derivatives, imaginary part, equations, real, eigenvalues, real function, Rayleigh quotient, vectors, critical points, eigenvectors, Hermitian matrix
	www.planetmath.org /encyclopedia/RayleighRitzTheorem.html (222 words)

	MGNet Preprints (Site not responding. Last check: 2007-10-16)
		The purpose of this paper is to develop a convergence theory for multigrid methods applied to nearly singular linear elliptic partial differential equations, of the type produced from a positive definite system by a shift with the identity.
		The theory is first applied to a method for computing eigenvalues and eigenvectors that consists of multigrid iterations with zero right-hand side and updating the shift from the Rayleigh quotient before every iteration.
		It is then applied to the Rayleigh quotient multigrid method (RQMG), which is a more direct multigrid procedure for solving eigenproblems.
	www.cerfacs.fr /~douglas/mgnet/papers/CaiZ-Mandel-McCormick/eig-dec93.html (110 words)

	MATH2071: LAB #9: The Eigenvalue Problem (Site not responding. Last check: 2007-10-16)
		, the Rayleigh quotient is a very accurate estimate of the corresponding eigenvalue.
		The Rayleigh quotient remains a valuable tool in the complex case, and most of the facts remain true.
		Plot the histories of Rayleigh quotient and first eigenvector component for all the cases.
	www.math.pitt.edu /~sussmanm/2071Spring05/lab09 (3735 words)

	MATH2071: LAB #12: The Eigenvalue Problem
		However, the preferred method for estimating the value of an eigenvalue uses the Rayleigh quotient.
		The Rayleigh quotient is a scalar value whose magnitude generally lies between the smallest and largest magnitudes of the eigenvalues of the matrix.
		When x is an approximate eigenvector of A, the Rayleigh quotient is a very accurate estimate of the corresponding eigenvalue (if real) or the magnitude of the eigenvalue (if complex).
	orion.math.iastate.edu /burkardt/math2071/lab_12.html (2014 words)

	Free Vibrations of Annular Plates Coupled with Fluids -- from Mathematica Information Center
		The accuracy of the assumed-modes approach is theoretically studied by using the Rayleigh-Ritz method that removes the simplifying hypothesis that dry and wet mode shapes are the same.
		Eigenfunctions of the plate vibrating in vacuum are assumed as admissible functions and the Rayleigh quotient for coupled vibration is used to obtain a Galerkin equation.
		It was found that the fundamental mode and frequency, for all the plate boundary conditions considered, is well estimated by the assumed-modes approach; higher modes are computed with less accuracy by this formulas and for some enhanced applications the Rayleigh-Ritz approach is necessary.
	library.wolfram.com /infocenter/Articles/1270 (213 words)

	Well-posedness and regularity properties of the Grassmann-Rayleigh quotient iteration
		Well-posedness and regularity properties of the Grassmann-Rayleigh quotient iteration
		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.
	www.montefiore.ulg.ac.be /~absil/Publi/GRQIextended.htm (74 words)

	3150quantum1.html (Site not responding. Last check: 2007-10-16)
		of any operator is the minimum of the Rayleigh quotient.
		Rayleigh quotient of A which is just the same as the Rayleigh
		quotient of H on the subspace of V spanned by the vectors psi[j].
	www.math.utah.edu:8080 /~mckay/3150quantum1.html (246 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-10-16)
		It can be shown that the Rayleigh quotient reaches its minimum value
		Specifically, this is the basis for Rayleigh quotient iteration.
		The Rayleigh quotient can be expressed as a function of the eigenvalues by decomposing any vector
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Rayleigh_quotient (199 words)

	Détail de publication -- Laboratoire de Mécanique de Lille
		Direct determination of the rotation in the polar decomposition of the deformation gradient by maximizing a Rayleigh quotient
		The maximization of the Rayleigh quotient is performed by a conjugate gradient algorithm with all iterative steps carried out by explicit closed formulae.
		Efficiency and accuracy of the method is illustrated by a numerical example.
	www.univ-lille1.fr /lml/?page=27&publID=271 (223 words)

	MATH2071: LAB #12: The Eigenvalue Problem
		Compute the value of the Rayleigh quotient for a variety of vectors x.
		What happens to the Rayleigh quotient when you look at a sequence of vectors x, Ax, AA*x,...
		The Rayleigh quotient R(A,x) will give us a way to approximate eigenvalues from an approximate eigenvector.
	people.scs.fsu.edu /~burkardt/math2071/lab_12.html (2014 words)

	Minimax principle (Site not responding. Last check: 2007-10-16)
		However the necessary absolute values entering the equations could be avoided, given an assurance that all the terms were real, as perhaps might be anticipated for the largest eigenvalue of a real matrix.
		It is generally understood that eigenvalues and eigenvectors satisfy a variational principle; for symmetric or hermitean matrices this leads to bounds involving the Rayleigh coefficient or to the the Courant minimax principle.
		If P were the set of coordinate vectors and Q were the set of vectors with positive components, the Rayleigh quotient would correspond to the right hand side of the Gerschgorin equations.
	delta.cs.cinvestav.mx /~mcintosh/oldweb/lcau/node98.html (466 words)

	slarrv.f
		This is crucial in the * case where bisection is used to ensure that the * eigenvalue is refined up to the required precision.
		SAVGAP = WGAP(WINDEX) WGAP(WINDEX) = GAP * We want to use the Rayleigh Quotient Correction * as often as possible since it converges quadratically * when we are close enough to the desired eigenvalue.
		* However, the Rayleigh Quotient can have the wrong sign * and lead us away from the desired eigenvalue.
	www.netlib.org /lapack/explore-html/slarrv.f.html (2381 words)

	Question about Rayleigh color matching test [Archive] - PPRuNe Forums
		I've taken a Rayleigh color matching test last week in order to check my color vision.
		the diagnosis was: Deutreanomalous tichromat, with a 0.92 anomalous quotient.
		I think you need to talk with an optometrist in your city (I think they are able to compare the Rayleigh result with the Nagel Anomaloscope)...
	www.pprune.org /forums/archive/index.php/t-210380.html (162 words)

	Criminal Justice Policy Foundation: Books: Principles and Techniques of Vibrations
		Only problem is the treatment of Rayleigh quotient.
		This book shows the Rayleigh quotient and its properties only for discrete systems but not for continuous systems.
		If this book reorganizes the topics in conservative systems versus nonconservative systems rather than discrete systems versus continuous systems, and this book is written in more general manner, it will be even more appreciated.
	www.cjpf.org /shop.php?mode=Books&item=0023801417 (294 words)

	Inverse iteration
		Once we have a suitable eigenvector approximation, we can use the Rayleigh quotient to find the eigenvalue.
		This is the idea behind Rayleigh quotient iteration.
		The inverse iteration algorithm requires solving a linear system at each step.
	www.xasa.com /wiki/en/wikipedia/i/in/inverse_iteration.html (234 words)

	Brian Hoffman's Publications (Site not responding. Last check: 2007-10-16)
		The fundamental postulate of quantum mechanics that starts one down the road to obtaining actual answers is the Average Value Theorem.
		In addition, the denominator of each quotient accounts for the fact that the wave functions might not be normalized and is equal to 1 if
		The quotient on the far right is written in the shorthand notation introduced by Dirac.
	zopyros.ccqc.uga.edu /~hoffbc/Source/scf/node3.html (801 words)

	details
		The basic idea is the same as the conjugate gradient (CG) method, i.e, minimizing the Rayleigh quotient via its gradient and avoiding reintroducing errors to the directions of previous gradients.
		Each iteration step is to find lowest eigenvector of the matrix in a subspace spanned by the current trial vector and the corresponding gradient of the Rayleigh quotient, as well as some previous trial vectors.
		The gradient, together with the previous trial vectors, play a similar role as the conjugate gradient of the original CG algorithm.
	cmbi.bjmu.edu.cn /ChinaPaper/details.asp?PubMedID=14682910 (159 words)

	JISE Vol.8 No.2 #8 (Site not responding. Last check: 2007-10-16)
		The Rayleigh quotient iterative algorithm with computational complexity O(N) was originally suggested for finding the eigenvalues of an N-by-N matrix with fast convergence rate.
		In this paper, a modified payleigh quotient iterative (MRQI) algorithm is proposed to track the desired roots, i.e.
		The MRQI algorithm associated with a zero suppression technique, called the parallel Rayleigh Quotient iterative (PRQI) algotithm, is further proposed to assure that rooting processors converge to different desired principal roots of spectral polynomials.
	www.iis.sinica.edu.tw /JISE/1992/199206_08.html (168 words)

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