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	Inverse iteration - Wikipedia, the free encyclopedia
		In numerical analysis, inverse iteration is an iterative eigenvalue algorithm.
		Whereas the power method always converges to the largest eigenvalue, inverse iteration also enables the choice of which eigenvalue to converge to.
		The inverse iteration algorithm requires solving a linear system at each step.
	en.wikipedia.org /wiki/Inverse_iteration (278 words)

	Power Iteration and Inverse Iteration (Site not responding. Last check: 2007-10-22)
		This module demonstrates power iteration and inverse iteration for computing an eigenvector corresponding, respectively, to the largest or smallest eigenvalue of a matrix.
		Normalized eigenvectors of the matrix are shown by arrows on the plot, with a dark arrow corresponding to the larger eigenvalue (in magnitude) and a light arrow corresponding to the smaller eigenvalue.
		The effect of each iteration is to tilt the starting vector closer to the dominant eigenvector (for power iteration) or subdominant eigenvector (for inverse iteration).
	www.cse.uiuc.edu /eot/modules/eigenvalues/EigenIteration (297 words)

	The following notes are to summarize (Site not responding. Last check: 2007-10-22)
		The convergence of inverse iteration is generally faster if we can choose the shift well.
		Reduction to tridiagonal form is advisable before using inverse iteration and a must before using the Rayleigh quotient method.
		It is based on inverse iteration, but that is hidden in the algorithm.
	www.cse.psu.edu /~barlow/cse456/eigen_notes.html (335 words)

	Seminar -- (Site not responding. Last check: 2007-10-22)
		Inexact inverse Iteration is inverse iteration where the arising shifted linear systems are not solved by direct methods like using an LU-decomposition but are approximated usually by using some sort of iterative technique.
		In contrast to inverse iteration which is a well known and well studied technique the convergence of inexact inverse iteration is not well established.
		As inexact inverse iteration is an inner outer type method the interest is not only in the convergence but also in how to choose stopping conditions and other parameters for the inner iterations to obtain a more efficient method.
	www-sccm.stanford.edu /seminar-w2005/Seminars_feb23_berns-mueller.htm (232 words)

	A Geometric Theory For Preconditioned Inverse Iteration Applied To A Subspace - Neymeyr (ResearchIndex)
		Abstract: The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix.
		The algorithm is built upon a subspace implementation of preconditioned inverse iteration, i.e.
		the well-known inverse iteration procedure, where the occurring system of linear equations is approximately solved by...
	citeseer.ist.psu.edu /neymeyr99geometric.html (583 words)

	Untitled Document (Site not responding. Last check: 2007-10-22)
		Inverse Iteration is a well known and well studied method for the calculation of A x =λ x, which requires solves with A-σ I where σ is the shift.
		A general convergence result is provided, which is independent of the linear solver and applicable to variation of inexact inverse iteration as proposed by several authors.
		Based on this convergence result and the convergence theory for inexact inverse iteration we provide a bound on the cost of an eigenvalue calculation.
	www.math.colostate.edu /~cruceanu/Teaching/Spring2003/Greenslopes/greensp03.html (1024 words)

	FRACTINT Inverse Julias (Site not responding. Last check: 2007-10-22)
		This method for drawing Julia Sets is called the Inverse Iteration Method, or IIM for short.
		Now, the inverse of Mandelbrot's classic function is a square root, and the square root actually has two solutions; one positive, one negative.
		Therefore at each step of each orbit of the inverse function there is a decision; whether to use the positive or the negative square root.
	spanky.triumf.ca /www/fractint/inv_julia_type.html (658 words)

	Julia_article.nb
		The random inverse iteration algorithm, also, highlights the similarities between Julia sets and self-similar sets, because of its resemblance to the chaos game [Devaney 1992, sec.
		Continuing, we iterate this procedure, choosing a random inverse and applying it to the previous point.
		The random inverse iteration algorithm may be improved, for some Julia sets, by skewing the probability of choosing one inverse over the other.
	www.unca.edu /~mcmcclur/professional/Julia/Links/index_lnk_7.html (330 words)

	Julia_article.nb (Site not responding. Last check: 2007-10-22)
		Inverse iteration algorithms are extremely fast, broadly applicable, and easily understood.
		Furthermore, an understanding of inverse iteration illuminates the connection between Julia sets of quadratic functions, more general rational functions and, even, self-similar sets.
		We then look at the implementation of inverse iteration for certain quadratic functions, carefully refine the technique, and generalize it to more arbitrary functions.
	www.unca.edu /~mcmcclur/professional/Julia/Links/index_lnk_1.html (155 words)

	sci.fractals FAQ (Site not responding. Last check: 2007-10-22)
		In inverse iteration, the equation z1 = z0^2 + c is reversed to give an equation for z0: z0 = ±sqrt(z1 - c).
		Iteration of this equation yields the period doubling route to chaos.
		Iterated function systems can be used to make things such as fractal ferns and trees and are also used in fractal image compression.
	www.faqs.org /faqs/sci/fractals-faq (12472 words)

	Eigenvalue algorithm
		By itself, power iteration is not very useful.
		Its convergence is slow except for special cases of matrices, and without modification, it can only find the largest or dominant eigenvalue (and the corresponding eigenvector).
		In addition, some of the better algorithms for the generalized eigenvalue problem are based on power iteration.
	www.brainyencyclopedia.com /encyclopedia/e/ei/eigenvalue_algorithm.html (378 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Inverse \ iteration algorithms are extremely fast, broadly applicable, and easily \ understood.
		Furthermore, an understanding of inverse iteration illuminates \ the connection between Julia sets of quadratic functions, more general \ rational functions and, even, self-similar sets.\n We begin with a brief \ look at the theory of Julia sets.
		The random inverse iteration algorithm may be improved, for some Julia \ sets, by skewing the probability of choosing one inverse over the other.
	www.cs.unca.edu /~mcclure/professional/Julia/Julia.nb (3073 words)

	Inverse Iteration (Site not responding. Last check: 2007-10-22)
		This method is called inverse iteration, or the inverse power method.
		One advantage of inverse iteration over the power method is the ability to converge to any desired eigenvalue (the one nearest
		This method is particularly effective when we have a good approximation to an eigenvalue and only want to compute this eigenvalue and its corresponding eigenvector.
	www.cs.utk.edu /~dongarra/etemplates/node96.html (165 words)

	Center for Computational Math Colloquium, 2000 (Site not responding. Last check: 2007-10-22)
		Preconditioned inverse iteration derives from inverse iteration (also called inverse power method) in a way that the occurring system of linear equations is solved approximately by using a preconditioner.
		The central result is a sharp convergence estimate for the Rayleigh quotient of the iterates.
		The convergence analysis of preconditioned inverse iteration is based on a predominantly geometric description: a central problem is to find the supremum of the Rayleigh quotient on a certain ball of iterates, which results from applying preconditioned inverse iteration for all admissible preconditioners to a fixed iterate.
	www-math.cudenver.edu /ccm/colloq/9900/mar27.html (278 words)

	NA01 Contributed Conference Abstracts (Site not responding. Last check: 2007-10-22)
		Inverse Iteration is a popular algorithm for the matrix eigenvalue problem with well understood convergence properties.
		in situations where LU decomposition can be used, and accurate approximations to the desired eigenvalues are known, inverse iteration is usually the method of choice, often enhanced by versions using subspaces.
		Further an analysis of strategies for parameter selection will be presented with the aim of reproducing the same asymptotic convergence rate of inverse iteration with exact linear solves.
	www.mcs.dundee.ac.uk /~naconf/01/abstracts/SUBbernsmueller.html (196 words)

	Johann Radon Institute for Computational and Applied Mathematics
		Inverse problems are concerned with determining (usually numerically) causes for desired or observed effects, which is a type of problems frequently arising in science and engineering.
		This means that a solution to an inverse problem might neither exist nor be unique, and even if some generalised concept of solution is introduced, then this solution depends in a discontinuous way on the data.
		When solving a complex inverse problems, e.g., a parameter identification problem for a transient and spatially three-dimensional partial differential equation, the efficient coupling of the "inverse iteration" with the "forward solver" (in this case, the method used for solving the pde with a given parameter) is crucial.
	www.ricam.oeaw.ac.at /research/invprob (471 words)

	inverse_iterations number_of_iterations (Site not responding. Last check: 2007-10-22)
		Number of iterations for inverse modeling of parameters.
		In each iteration the total time history as defined by the control_timestep records will be applied.
		At the end of the iteration, these sensitivities are used to determine new estimates for the parameters.
	tochnog.sourceforge.net /tnu/node374.html (79 words)

	Sexual Paradox: Chaos
		As the growth rate varies, the iteration goes through a sequence of different stages separated by sudden changes, or bifurcations.
		where W is a constant twist by a fixed angle per iteration and the sine term causes a wobbling or pumping effect, resulting in an average twisting which may become mode-locked.
		Top left: Julia set plotted by inverse iteration showing the multifractal distribution of how frequently differing subregions are visited.
	www.dhushara.com /paradoxhtm/chaos.htm (5380 words)

	[No title]
		Since the map F is analytic (and therefore continuous), the ordering of the above inverse iterates zN,j on Jc should correspond to the ordering of the first 2N inverse iterates under z2 of the fixed point 1 on the unit circle.
		At the kth inverse iteration step, we have 2k points on the unit circle: {zk,0,...
		Geometrically speaking, the “positive” inverse iterates are obtained by halving the polar coordinate angles of each zk,j.
	www.math.psu.edu /mazzucat/VIGRE/Sushrut_Gautam.doc (1052 words)

	IMACS03 conference abstract: Matsekh (Site not responding. Last check: 2007-10-22)
		To overcome shortcomings of both Godunov's method and Inverse Iteration (often nondeterministic character of starting vectors, need to introduce disturbances into the shift, high worst case complexity) we constructed a new hybrid procedure -- the Godunov-Inverse Iteration.
		Godunov-Inverse Iteration is very robust with respect to the choice of the Inverse Iteration shift -- we use right-hand bounds of the eigenvalue intervals computed by the bisection method as extremely accurate shifts in the Godunov-Inverse Iteration.
		As a result Godunov-Inverse Iteration produces accurate and robust solutions to the symmetric eigenvalue problem with higher accuracy than Godunov's method and in fewer steps than existing implementations of the Inverse Iteration.
	www-math.cudenver.edu /IMACS03/abs/matsekh.html (430 words)

	Computing an Eigenvector with Inverse Iteration
		The purpose of this paper is two-fold: to analyze the behavior of inverse iteration for computing a single eigenvector of a complex square matrix and to review Jim Wilkinson's contributions to the development of the method.
		We also explain the often significant regress of the residuals after the first iteration: it occurs when the non-normal part of the matrix is large compared to the eigenvalues of smallest magnitude.
		We conclude that the behavior of the residuals in inverse iteration is governed by the departure of the matrix from normality rather than by the conditioning of a Jordan basis or the defectiveness of eigenvalues.
	epubs.siam.org /sam-bin/dbq/article/30077 (271 words)

	Adaptive Inner-Outer Inverse Iteration for Eigenvector Computations (ResearchIndex)
		Abstract: Inverse iteration is a standard technique for finding selected eigenvectors associated with eigenvalues which are known approximately.
		1.8: Adaptive Inner-Outer Inverse - Iteration For Eigenvector (1998)
		6 An inexact inverse iteration for large sparse eigenvalue pro..
	citeseer.ist.psu.edu /223206.html (337 words)

	Joerg Berns-Mueller
		One effective and well known and well studied method for calculating a simple eigenvalue and its corresponding eigenvector is inverse iteration.
		Thereby we allow several generalisations of inverse iteration, for example, solving correction equations ore more importantly allowing to use a modified right hand side.
		In case of symmetric eigenvalue problem the formulation of inexact inverse iteration where a modified right hand side is used terns out to be most efficient.
	www.math.uni-frankfurt.de /~berns/Eigenwerte (424 words)

	lapack-z/zhsein.html
		This property allows ZHSEIN to perform inverse iteration on just one diagonal block.
		In this case, ZHSEIN must always perform inverse iteration using the whole matrix H. INITV (input) CHARACTER*1 = 'N': no initial vectors are supplied; = 'U': user-supplied initial vectors are stored in the arrays VL and/or VR.
		W (input/output) COMPLEX*16 array, dimension (N) On entry, the eigenvalues of H. On exit, the real parts of W may have been altered since close eigen- values are perturbed slightly in searching for independent eigenvectors.
	www.math.utah.edu /software/lapack/lapack-z/zhsein.html (624 words)

	Matrix eigenvalue problem (Site not responding. Last check: 2007-10-22)
		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)

	Abstract for Alastair Spence Seminar (Site not responding. Last check: 2007-10-22)
		Often only a small number of eigenvalues will be required, for example in stability analysis when initial estimates for a shift may also be available.
		Inverse iteration is just the power method applied to the shift-invert transformation.
		0, then the whole algorithm may be thought of as an ``inner-outer'' iteration applied to the shift-invert transformation.
	www.brunel.ac.uk /~mastsnc/seminars/semsum114.html (145 words)

	[No title]
		The power iteration method is a very well-known tool for approximating the largest eigenvalue of a symmetric, positive definite matrix.
		Although being a close relative of the conjugate gradient iteration for solving linear systems of equations, the convergence theory for the Lanczos method is less developed.
		Note that this is worse than the Rayleigh quotient iteration (cf., e.g., [10]), which is known to have a cubic convergence rate locally.
	www.ubka.uni-karlsruhe.de /vvv/1997/mathematik/10/10.text (3413 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		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...
		It can be proved that R(ayleigh)-shift and W(ilkinson)-shift are approximations of at least one eigenvalue of A ; in that sense, shifted inverse iteration is also a method to find eigenvalues.
		If A is symmetric, it actually happens to be the best one known (since QR, LR and related approaches can be viewed as pertinent and computationnaly reliable implementations of inverse iteration).
	www.math.niu.edu /~rusin/known-math/01_incoming/rayleigh (402 words)

	invit.f
		C C On INPUT C C NM must be set to the row dimension of the two-dimensional C array parameters, A and Z, as declared in the calling C program dimension statement.
		C This array holds the triangularized form of the upper C Hessenberg matrix used in the inverse iteration process.
		They hold the approximate eigenvectors during the C inverse iteration process.
	www.cs.yorku.ca /~roumani/fortran/slatecAPI/invit.f.html (666 words)

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