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Topic: Eigenvalue algorithm

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Eigenvalue

Companion matrix

Hessenberg matrix

QR algorithm

Condition number

QR decomposition

Numerical analysis

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	Divide-and-conquer eigenvalue algorithm - Wikipedia, the free encyclopedia
		Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s) become competitive in terms of stability and efficiency with more traditional algorithms such as the QR algorithm.
		An eigenvalue problem is divided into two problems of roughly half the size, each of these are solved recursively, and the eigenvalues of the original problem are computed from the results of these smaller problems.
		As with most eigenvalue algorithms for Hermitian matrices, divide-and-conquer begins with a reduction to tridiagonal form.
	en.wikipedia.org /wiki/Divide-and-conquer_eigenvalue_algorithm (910 words)

	Eigenvalue algorithm - Wikipedia, the free encyclopedia
		Therefore, general eigenvalue algorithms are expected to be iterative.
		In addition, some of the better algorithms for the generalized eigenvalue problem are based on power iteration.
		A popular method for finding eigenvalues is the QR algorithm, which is based on the QR decomposition.
	en.wikipedia.org /wiki/Eigenvalue_algorithm (505 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Algorithm Description The general problem is that given a symmetric positive definite matrix A, find those values alpha and those vectors x such that: A * x = alpha * x (1) where the first multiplication is a matrix-vector multiplication and the second multiplication is a scalar-vector multiplication.
		A'n such that A'n is close to diagonal and has the same eigenvalues as the original A. Ten years ago, I implemented this algorithm in Fortran for the Cray-1, this function is a C implementation of this same algorithm.
		The algorithm assumes the relevant part of A is stored in the upper right triangle of A, the lower left and diagonal elements are not changed.
	mywebpages.comcast.net /mike_ess/jacobi.txt (863 words)

	List of algorithms
		See also the list of data structures, list of algorithm general topics and list of terms relating to algorithms and data structures.
		Snapshot algorithm: a snapshot is the process of recording the global state of a system
		Rainflow-counting algorithm: Reduces a complex stress history to a count of elementary stress-reversals for use in fatigue analysis
	starrepublic.org /encyclopedia/wikipedia/l/li/list_of_algorithms.html (779 words)

	Cycle Versus Cumulative Fission Matrix Algorithm
		The dominant eigenvalue, or K, of the fission matrix A is evaluated numerically.
		From this figure we notice that the batch eigenvalue of the cumulative fission matrix is far superior to the batch eigenvalue of the cycle fission matrix, as expected, whereas the batch averaged eigenvalue of the cycle fission matrix and the batch eigenvalue of the cumulative fission matrix show comparable accuracy.
		The batch eigenvalue of the cumulative fission matrix algorithm and the batch averaged eigenvalue of the cycle fission matrix algorithm provide comparable results for tightly coupled problems, where there is enough neutron communication between different regions.
	www.sdsc.edu /~majumdar/thesis/node22.html (985 words)

	Multiplication Algorithm
		Hence, there are pos of eigenvalues equal to d and neg of eigenvalues equal to -d, for a total of m with a magnitude of d.
		This multiplication algorithm is equivalent to the Horner's method for finding the roots of a polynomial.
		Deflation in the case of the multiplication algorithm for a matrix corresponds to the division by the found factor in the Horner's method for a polynomial.
	www.rism.com /LinAlg/multiplication_algorithm.htm (3162 words)

	Eigenvalue, eigenvector and eigenspace - (Site not responding. Last check: 2007-10-08)
		This is the characteristic polynomial of A: the eigenvalues of a matrix are the zeros of its characteristic polynomial.
		The algebraic multiplicity of an eigenvalue λ of A is the order of λ as a zero of the characteristic polynomial of A; in other words, if λ is one root of the polynomial, it is the number of factors (t − λ) in the characteristic polynomial after factorization.
		In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix A, or (increasingly) of the graph's Laplacian matrix $I-T^{-1/2}AT^{-1/2}$ , where T is a diagonal matrix holding the degree of each vertex, and in $T^{-1/2}$ , 0 is substituted for $0^{-1/2}$ .
	psychcentral.com /psypsych/Eigenvector (4313 words)

	Eigenvalue Problem
		One of the oldest and most general approaches for the solution of the classical eigenvalue problem is the Jacobi method that was first presented in 1846.
		The classical Jacobi eigenvalue algorithm is summarized within the computer subroutine given in Table D.1.
		The Jacobi algorithm can be directly applied to all off-diagonal terms, in sequence, until all terms are reduced to a small number compared to the absolute value of all terms in the matrix.
	www.csiberkeley.com /Tech_Info/eigenProblem.htm (852 words)

	List of algorithms - Wikipedia, the free encyclopedia
		Line drawing algorithm: graphical algorithm for approximating a line segment on discrete graphical media.
		Buddy memory allocation: Algorithm to allocate memory such that fragmentation is less.
		LR parser: A more complex linear time parsing algorithm for a larger class of context-free grammars.
	www.sciencedaily.com /encyclopedia/list_of_algorithms (1659 words)

	Eigenvalue Problems (Site not responding. Last check: 2007-10-08)
		Eigenvalue problems have also provided a fertile ground for the development of higher performance algorithms.
		These algorithms generally all consist of three phases: (1) reduction of the original dense matrix to a condensed form by orthogonal transformations, (2) solution of condensed form, and (3) optional backtransformation of the solution of the condensed form to the solution of the original matrix.
		The first step in solving many types of eigenvalue problems is to reduce the original matrix to a condensed form by orthogonal transformations.
	www.netlib.no /netlib/lapack/lug/node70.html (1281 words)

	Introduction
		In general, the larger or more difficult an eigenvalue problem, the more important it is to use an algorithm that exploits as much of its mathematical structure as possible (such as symmetry or sparsity).
		An eigenvalue or eigenspace is called well-conditioned if its error bound is acceptably small for the user (this obviously depends on the user), and ill-conditioned if it is much larger.
		Chapter 10 treats data structures, algorithms, and software for sparse matrices, especially sparse linear solvers, which often are the most time-consuming part of an eigenvalue algorithm.
	www.cs.utk.edu /~dongarra/etemplates/node19.html (1025 words)

	[No title]
		With the exception of this abstract, the text of this document is the index to the TOMS library which was requested from and was received from NETLIB on June 4, 1993.
		Dissemination Agreement The ACM Algorithms Policy says, in part, Dissemination Agreement Submittal of an algorithm for publication in one of the ACM Transactions implies that unrestricted use of the algorithm within a computer is permissible.
		General permission to copy and distribute the algorithm without fee is granted provided that the copies are not made or distributed for direct commercial advantage.
	www.uic.edu /depts/adn/infwww/txt/v2808001.txt (1537 words)

	The BR Eigenvalue Algorithm - Geist, Howell, Watkins (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced.
		It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process.
		SR and SZ Algorithms for the Symplectic (Butterfly)..
	citeseer.ist.psu.edu /407317.html (410 words)

	Monte Carlo K algorithms
		The source iteration and fission matrix algorithms for Monte Carlo criticality calculations are discussed below using the collision estimator.
		is a random number between 0 and 1, and K is the previous batch eigenvalue (or a guessed one for the first batch).
		The dominant eigenvalue, or K, of the fission matrix A can be calculated by using a matrix iterative algorithm.
	www.sdsc.edu /~majumdar/thesis/node16.html (772 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Regularity results are proved for the eigenfunctions of such problems; error estimates are given for the approximation of the eigenvalues and eigenfunctions using the Rayleigh-Ritz method, and a detailed analysis is made of numerical results obtained by applying finite element methods to a model problem.
		The new estimates obtained for the error between a multiple eigenvalue and the closest approximate eigenvalue depend on only one eigenvector from the space of corresponding eigenvectors, namely the one best approximated by the space employed in the Galerkin method.
		The approximation properties of the approximate eigenvalue which is farthest from a given multiple eigenvalue are also studied.
	www-math.cudenver.edu /~aknyazev/research/papers/andrew.bib (3916 words)

	[No title]
		Adaptive mesh algorithms communicate information about the numerical solution between levels of the hierarchy and also among grids at the same level of the hierarchy.
		We present computational results for the calculation of the lowest eigenvalue and associated eigenvector of the 3d Hamiltonian for a ring of ten hydrogen ions located in the Z=0 plane.
		We divide the execution time for the eigenvalue algorithm, including time spent building the adaptive grid hierarchy, into numerical computation, time lost to load imbalance, communication among grids at the same level of the hierarchy, communication between levels, error estimation, load balancing, and grid generation.
	www-cse.ucsd.edu /groups/hpcl/scg/papers/1996/amg_sc95/sc95.txt (5833 words)

	Luigi Dragone Home Page - Spectral Clusterer for WEKA
		In [3] is presented another version of spectral clustering algorithm, and a comparative analysis of various clustering algorithms based on spectral methods is in [4].
		The implemented algorithm is formulated as graph partition problem where the weight of each edge is the similarity between points that correspond to vertex connected by the edge.
		The goal of the algorithm is find the minimum weight cuts in the graph, but this problem can be addressed by the means of linear algebra, in particular by the eigenvalue decomposition techniques, from which the term "spectral" derives.
	www.luigidragone.com /datamining/spectral-clustering.html (1086 words)

	ASAP '99 Principal-Components, Covariance Matrix Tapers... - Joseph R. Guerci and Jameson S. Bergin (Site not responding. Last check: 2007-10-08)
		Unfortunately this advantage is essentially lost in those situations were eigenvalue spreading occurs-a problem that appears evident in many real-world data sets and is justifiable based on physical modeling of the environments.
		In this paper, a new approach to adaptive interference mitigation is presented based on the recognition that many eigenvalue spreading mechanisms (e.g., internal clutter motion, clutter scintillation, and jammer/clutter diffuse multipath) are effectively modeled as a covariance matrix tapering (CMT) of the interference covariance matrix associated with the "unspread" dominant eigenvalues.
		For modest spreading environments, restoration is readily achieved by first estimating the covariance matrix associated with the K dominant eigenvectors, then applying a CMT to account for the remaining eigenvalues.
	www.ll.mit.edu /asap/asap_99/abstract/8.html (290 words)

	Research topics, Eigenvalues
		In this department, a large effort was done in understanding and developing robust numerical methods for the calculation of the rightmost eigenvalue of large sparse matrices.
		Research was concentrated on the issue of implicitly restarting iterative eigenvalue algorithm, especially the Rational Krylov Sequence method and the unsymmetric Lanczos algorithm.
		Analogously, the possibility of implicitly restarting and implicitly filtering inexact eigenvalue solvers is studied.
	www.cs.kuleuven.ac.be /~nalag/research/topics/eigenval.shtml (176 words)

	TRLAN User Guide
		The underlying algorithm of TRLAN is a dynamic thick-restart Lanczos algorithm.
		For symmetric eigenvalue problems, the residual norm is one of the most commonly used measure of the solution accuracy.
		The shift-and-invert scheme computes the extreme eigenvalues of first, then derive the actual eigenvalues of To use this scheme, one need to either invert the matrix or at least being able to solve linear systems, If neither is feasible, then the Davidson method might be an alternative to consider.
	crd.lbl.gov /~kewu/ps/trlan_.html (7254 words)

	Effective dimensional reduction algorithm for eigenvalue problems for thin elastic structures: A paradigm in three ...
		Effective dimensional reduction algorithm for eigenvalue problems for thin elastic structures: A paradigm in three dimensions -- Ovtchinnikov and Xanthis 97 (3): 967 -- Proceedings of the National Academy of Sciences
		the convergence of this algorithm obtained from that of ref. 23.
		From 11 we observe that the convergence of the algorithm 9 is affected by two factors: (i) the spectral condition
	www.pnas.org /cgi/content/full/97/3/967 (2765 words)

	G03 Manual: OPT
		For the Hartree-Fock, CIS, MP2, MP3, MP4(SDQ), CID, CISD, CCD, CCSD, QCISD, CASSCF, and all DFT and semi-empirical methods, the default algorithm for both minimizations (optimizations to a local minimum) and optimizations to transition states and higher-order saddle points is the Berny algorithm using redundant internal coordinates [149,15] (specified by the Redundant option).
		This was sometimes superior to the Berny method in Gaussian 88, but since the RFO step [530] has now been incorporated into the Berny algorithm, EF is seldom preferable unless its ability to follow a particular mode is needed, or gradients are not available (in which case Berny can't be used anyway).
		The Berny geometry optimization algorithm in Gaussian is based on an earlier program written by H. Schlegel which implemented his published algorithm [136].
	www.gaussian.com /g_ur/k_opt.htm (8085 words)

	Watkins, Refereed Publications
		The transmission of shifts and shift blurring in the QR algorithm (.ps), Linear Algebra Appl., 241-243 (1996), pp.
		SR and SZ algorithms for the symplectic (butterfly) eigenproblem (.ps), with P. Benner and H. Fassbender, Linear Algebra Appl., 287 (1999), pp.
		The BR eigenvalue algorithm (.ps), with G. Geist and G. Howell, SIAM J. Matrix Anal.
	www.sci.wsu.edu /math/faculty/watkins/refpub.html (776 words)

	About Horst D. Simon: Curriculum Vitae
		His research interests are in the development of sparse matrix algorithms, algorithms for large-scale eigenvalue problems, and domain decomposition algorithms for unstructured domains for parallel processing.
		Horst's recursive spectral bisection algorithm is regarded as a breakthrough in parallel algorithms for unstructured computations, and his algorithm research efforts were honored with the 1988 Gordon Bell Prize for parallel processing research.
		Estimating the Largest Eigenvalue of a Symmetric Positive Definite Matrix with the Lanczos Algorithm” (with B. Parlett and L. Stringer), Mathematics of Computation 38, 153-165, 1982.
	www.nersc.gov /~simon/about/cv.html (8678 words)

	Eigensolve (Site not responding. Last check: 2007-10-08)
		The algorithm is described in the paper An iterated eigenvalue algorithm for approximating the roots of univariate polynomials (an earlier
		This computation required 7 eigenvalue computations and took about 87 seconds (on a 250 Mhz SGI R10000, with roots reported to 15 decimal digits).
		Here is a description of the algorithm and other examples of iterations.
	cm.bell-labs.com /who/sjf/eigensolve.html (247 words)

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