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Topic: Matrix eigenvalue problem

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	Encyclopedia :: encyclopedia : Eigenvalue (Site not responding. Last check: 2007-10-30)
		The solution to the eigenvalue equation is N = exp(λt), the exponential function; thus that function is an eigenfunction of the differential operator d/dt with the eigenvalue λ.
		This is the characteristic polynomial of A: the eigenvalues of a matrix are the zeros of its characteristic polynomial.
		The eigenvalues of a triangular matrix are the entries on the main diagonal.
	www.hallencyclopedia.com /Eigenvalue (4604 words)

	PlanetMath: eigenvalue problem
		Many problems in physics and elsewhere lead to differential eigenvalue problems, that is, problems where the vector space is some space of differentiable functions and where the linear operator involves multiplication by functions and taking derivatives.
		Matrix eigenvalue problems arise in a number of different situations.
		This is version 18 of eigenvalue problem, born on 2002-01-14, modified 2006-06-15.
	planetmath.org /encyclopedia/EigenvalueProblem.html (520 words)

	Computational physics - Wikipedia, the free encyclopedia
		Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists.
		Computational physics borrows a number of ideas from computational chemistry - for example, the density functional theory used by computational physicists to calculate properties of solids is basically the same as that used by chemists to calculate the properties of molecules.
		Many other more general numerical problems fall loosely under the domain of computational physics, although they could easily be considered pure mathematics or part of any number of applied areas.
	en.wikipedia.org /wiki/Computational_physics (281 words)

	A Matrix Eigenvalue Problem (Site not responding. Last check: 2007-10-30)
		The following eigenvalue problem was sent to me by Professor J. Luttinger, a theoretical physicist at Columbia.
		The problem is concerned with the representation of the largest eigenvalue of a class of matrices.
		The largest eigenvalue is represented as an integral (from zero to $2\pi$) of the log of a trigonometric polynomial.
	www.csc.fi /math_topics/Mail/NANET94/msg00041.html (132 words)

	Matrix Eigenvalue Problem of Multiple-Shaker Testing (Site not responding. Last check: 2007-10-30)
		It is shown that the problem of dynamic testing using multiple shaker excitations can be represented by a matrix-eigenvalue formulation.
		The normalized eigenvector corresponding to the maximum eigenvalue provides the direction of excitation for an equivalent single shaker.
		The corresponding optimal intensity of excitation is expressed in terms of the eigenvalues.
	www.pubs.asce.org /WWWdisplay.cgi?8200362 (70 words)

	Matrix Set CRYSTAL (Site not responding. Last check: 2007-10-30)
		This is a large nonsymmetric standard eigenvalue problem that arises from the stability analysis of a crystal growth problem.
		The matrix eigenvalue problem follows from discretization using the standard second order finite difference formulae.
		The Matrix Market is a service of the Mathematical and Computational Sciences Division / Information Technology Laboratory / National Institute of Standards and Technology.
	math.nist.gov /MatrixMarket/data/NEP/crystal/crystal.html (221 words)

	Branch-and-Cut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem - Fukuda (ResearchIndex) (Site not responding. Last check: 2007-10-30)
		Branch-and-Cut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem (1999)
		Abstract: The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of the control and system theory in the last few years.
		However, on the contrary of the Linear Matrix Inequality (LMI) which can be solved by interior-point-methods, the BMI is a computationally difficult object in theory and in practice.
	citeseer.ist.psu.edu /204561.html (687 words)

	Lapack testing problems (Site not responding. Last check: 2007-10-30)
		Matrix order= 5, type=19, seed=1473,2247,3104,3869, result 1 is 1.600E+05 Matrix order= 5, type=23, seed= 98, 522, 225, 169, result 1 is 1.760E+05 Matrix order= 5, type=25, seed=1468,2085,3970, 617, result 1 is 1.063E+05 CDRVGG: CGEGV returned INFO= 6.
		Matrix order= 5, type=20, seed= 781,1305,2295, 317, result 1 is 8.389E+06 Matrix order= 5, type=24, seed= 712,3117, 514, 249, result 1 is 5.605E+04 CDRVGG: CGEGV returned INFO= 5.
		Matrix order= 5, type=17, seed=1915,2136,2450, 53, result 5 is 4.391E+13 Matrix order= 5, type=17, seed=1915,2136,2450, 53, result 12 is 8.535E+13 Matrix order= 5, type=20, seed=2617, 913,1978,1125, result 5 is 1.082E+11 Matrix order= 5, type=20, seed=2617, 913,1978,1125, result 12 is 3.135E+11 ZCHKGG: ZHGEQZ(S) returned INFO= 5.
	www.scyld.com /pipermail/beowulf/2003-January/009245.html (8366 words)

	Pseudospectra & Structural Dynamics
		The determination of natural frequencies and mode shapes require the solution of an eigenvalue problem, which in turn depends on the material properties, geometric parameters and boundary conditions of the structure under consideration.
		The ultimate goal of studying random eigenvalue problems is to obtain the joint probability density function of the eigenvalues and the eigenvectors.
		The problem of non-convergence is due to the presence of the interface and the corresponding interfacial mode.
	eis.bris.ac.uk /~enxtw/abstracts.html (2270 words)

	MTHSC 434/634: Outlines
		We will look for the common thread in three apparently dissimilar problems: finding the distance from a point to a plane, modeling noisy data, and estimating the temperature profile in a heated rod.
		Looked at an approximating problem which turns out to be a matrix eigenvalue problem.
		We solved an eigenvalue problem for the symmetric matric A = [2 1 1; 1 2 1; 1 1 2].
	www.math.clemson.edu /~bmoss/434/outlines (608 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #6313082 - Matrix inverse eigenvalue problem for periodic Jacobi matrices
		Matrix inverse eigenvalue problem for periodic Jacobi matrices
		A stable numerical algorithm is presented for generating a periodic Jacobi matrix from two sets of eigenvalues and the product of the off-diagonal elements of the matrix.^The algorithm requires a simple generalization of the Lanczos algorithm.^It is shown that the matrix is not unique, but the algorithm will generate all possible solutions.^3 references.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=6313082 (165 words)

	Preconditioned iterative methods for monotone nonlinear eigenvalue problems
		This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of the symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter.
		The discretization of these eigenvalue problems leads to ill-conditioned nonlinear eigenvalue problems with very large sparse matrices monotone depending on the spectral parameter.
		To compute the smallest eigenvalue of large matrix nonlinear eigenvalue problem, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method.
	archiv.tu-chemnitz.de /pub/2006/0065 (161 words)

	Moving Least-Squares and Wavelets for Random Field Characterization (Site not responding. Last check: 2007-10-30)
		Two numerical methods were developed for solving the integral eigenvalue problem associated with K-L expansion.
		In the first method, the eigenfunctions were approximated as linear sums of wavelets and the integral eigenvalue problem was converted to a finite-dimensional matrix eigenvalue problem that can be easily solved.
		In the second method, a Galerkin-based approach in conjunction with meshless discretization was developed in which the integral eigenvalue problem was also converted to a matrix eigenvalue problem.
	www.ccad.uiowa.edu /~rahman/nsfcareer/mlawfrfc.html (245 words)

	Computational physics (Site not responding. Last check: 2007-10-30)
		Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which aquantitative theory already exists.
		Unfortunately, it is often thecase that solving the theory's equations in order to produce a useful prediction is a computationally difficult problem.
		Many other more general numerical problems fall loosely under the domain of computational physics, although they could easilybe considered pure mathematics or part of any number of applied areas.
	www.therfcc.org /computational-physics-56.html (201 words)

	Solving the Discretized Problem (Site not responding. Last check: 2007-10-30)
		Finally, once the eigenvalues are found, the eigenvectors are computed using inverse iteration.
		This inverse iteration technique is described in numerous texts on algebraic eigenvalue problems, see for instance, Wilkinson [62].
		Unfortunately, even though the eigenvalues of a purely acoustic problem are guaranteed to be distinct, they can be very nearly degenerate.
	hlsresearch.com /oalib/Modes/AcousticsToolbox/manual_html/node38.html (321 words)

	Computational physics - One Language (Site not responding. Last check: 2007-10-30)
		Physicists often have a very precise mathematical theory describing how a system will behave.
		Unfortunately, it is often the case that solving the theory's equations in order to produce a useful prediction is a computationally difficult problem.
		In addition, the computational cost of solving quantum mechanical problems is generally exponential in the size of the system (see computational complexity theory).
	www.onelang.com /encyclopedia/index.php/Computational_physics (237 words)

	On matrix inverse eigenvalue problems
		It is proven that this problem is equivalent to the joint eigenpair problem of a family of related matrix pencils.
		Furthermore, the conditions under which this problem possesses a solution are given.
		A method for solving the joint eigenpair problem of a family of matrix pencils is then presented, which, in turn, gives all solutions of the problem.
	stacks.iop.org /0266-5611/14/275 (232 words)

	Matrix Generator MVMODE
		Consider the following eigenvalue problem of an ordinary differential equation
		Replacing the second derivatives in (3) with a centered difference operators to obtain the generalized matrix eigenvalue problem
		Problem (4) can be recast as the standard eigenvalue problem
	math.nist.gov /MatrixMarket/data/NEP/mvmode/mvmode.html (181 words)

	NA Digest, V. 94, # 10
		Hence, those are all the eigenvalues of A, with eigenvectors v[1],,...v[k].
		The eigenvalue problem for T can be treated in a similar way as in
		Eigenvalues of Block Matrices arising from Discretizations of the Navier
	www.netlib.org /na-digest-html/94/v94n10.html (3298 words)

	Mathematics Discipline Staff - Dr Nian Li
		Li, N., An Inverse Eigenvalue Problem for Block Matrices, WSEAS Transactions on Computers, 3, Vol 2, 2003, 824-828.
		Li, N. and Steiner, J.,A new algorithm for solving inverse eigenvalue problems, Acta Technica Acad Sci Hung 108 (1-2), (1999) 143-148.
		Li, N., A matrix inverse eigenvalue problem and its application, Linear Algebra Appl.
	www.swin.edu.au /feis/mathematics/staff/nli.html (174 words)

	Yuji YAMADA
		Yamada and S. Hara, “The Matrix Product Eigenvalue Problem: Global optimization for the spectral radius of a matrix product under convex constraints,” Proceedings of the IEEE Conference on Decision and Control, pp.
		Yamada and S. Hara, “Global Optimization for the Matrix Produce Eigenvalue Problem,” The 27th SICE Symposium on Control Theory, Hanamaki, Japan, 1998.
		Yamada and S. Hara, “Global optimization of a matrix product eigenvalue under LMI constraints with monotonicity property,” The 26th SICE Symposium on Control Theory, Chiba, Japan, 1997.
	www.cds.caltech.edu /~yuji/pub.html (608 words)

	Atlas: On a Parallel Maxwell Eigensolver by Peter Arbenz (Site not responding. Last check: 2007-10-30)
		We found the Jacobi-Davidson (JD) method to be a very effective factorization-free algorithm to compute 5-10 eigenvalues at the low end of the spectrum of (1) provided that a good preconditioner is available for the correction equations that have to be solved in each step of the JD iteration.
		It is close to optimal regarding iteration count and scales with regard to memory consumption.
		Multilevel preconditioners for solving eigenvalue problems occuring in the design of resonant cavities.
	atlas-conferences.com /cgi-bin/abstract/caol-08 (317 words)

	Distributing Matrix Eigenvalue Calculations over Transputer Arrays (ResearchIndex) (Site not responding. Last check: 2007-10-30)
		Abstract: We discuss the parallel numerical solution of the matrix eigenvalue problem for real symmetric tridiagonal matrices.
		617 The Algebraic Eigenvalue Problem (context) - Wilkinson - 1965
		1 Solution of eigenvalue problems in occam and transputers (context) - Ralha, Thomas - 1988
	citeseer.ist.psu.edu /495685.html (309 words)

	ScienceDaily: Algorithm Advance Produces Quantum Calculation Record
		To make the problem computationally practical, Sims and Hagstrom merged two earlier algorithms for these calculations--one which has advantages in ease of calculation, and one which more rapidly achieves accurate results--into a hybrid with some of the advantages of both.
		The final calculations were run on a 147-processor parallel cluster at NIST over the course of a weekend--on a single processor it would have taken close to six months.
		Solutions to problems in the field of digital image processing generally require extensive experimental work involving software simulation and testing with large sets of sample images.
	www.sciencedaily.com /releases/2006/03/060320215143.htm (1714 words)

	[No title]
		Appl.", year = "1995", volume = "2", number = "ü3", pages = "195-203", keywords = "{J}acobi matrix, arrow matrix", source = "Xu" } @incollection {bosz81, author = {Bosznay, {\'A}dam}, title = {Solution of the inverse eigenvalue problem of a vibrating continuum with the method of intermediate operators}, BOOKtitle = {Numerical treatment of differential equations, Vol.
		{A}pplication to the inverse eigenvalue problem}, BOOKtitle = {Proceedings of the First Conference of Portuguese and Spanish Mathematicians (Lisbon, 1972) (Spanish)}, pages = {344--351}, PUBLISHER = {Inst.
		Heged{\H{u}}s}, } @article {deak92, author = {Deakin, A. and Luke, T. title = {On the inverse eigenvalue problem for matrices}, journal = {J. Phys.
	www4.ncsu.edu /~mtchu/Research/Lectures/Iep/ref_main.bib (3583 words)

	Imtroduction to Molecular Modeling Class Wed. and Fri. Sept. 8 &10
		It first arose in astronomical calculations about the long-term behavior of planetary orbits.
		If we omit S then it is HC=EC a matrix eigenvalue problem.
		•Mathematicians have developed efficient algorithms for solving both kinds of matrix eigenvalue problem.
	uark.edu /campus-resources/pulay/Modeling04/class1_files/slide0049.htm (88 words)

	Computational physics Article, Computationalphysics Information (Site not responding. Last check: 2007-10-30)
		The matrix eigenvalue problem – i.e.the problem of finding eigenvalues of very large matrices.
		comptational physics, theory, computational phyics, problem, computational physisc, system, computational phyiscs, quantum, computatioanl physics, method, computatioal physics, numerical, compuational physics, number, omputational physics, part, computationa physics, differential, computationla physics, mechanical,, mathematics, computatoinal physics, size, computational phisics, complexity, computationl physics, loosely, computational physcs, many, computational hysics, evaluating, computtional physics, specifically, computational physcis, matrix, copmutational physics, methodand,...
		We take no responsibility for the content, accuracy and use of this article.
	www.anoca.org /problems/theory/computational_physics.html (282 words)

	Numerical Solution of the Spinless Salpeter Equation by a Semianalytical Matrix Method (A Mathematica 4.0 Routine) -- ...
		Numerical Solution of the Spinless Salpeter Equation by a Semianalytical Matrix Method (A Mathematica 4.0 Routine)
		In quantum theory, the "spinless Salpeter equation", the relativistic generalization of the nonrelativistic Schrödinger equation, is used to describe both the bound states of scalar particles and the spin-averaged spectra of bound states of fermions.
		A numerical procedure solves the spinless Salpeter equation by approximating this eigenvalue equation by a matrix eigenvalue problem with explicitly known matrices.
	library.wolfram.com /infocenter/Articles/2496 (110 words)

	CS339: Eigenvalue Computations (Site not responding. Last check: 2007-10-30)
		This course covers numerical methods for solving the matrix eigenvalue problem.
		Related problems such as the singular value decomposition and generalized eigenvalue problems will be discussed, as well as applications such as the solution of polynomial equations, orthogonal polynomials, and numerical quadrature.
		References will be listed on the References page and/or distributed in class.
	www.stanford.edu /class/cs339/course.html (76 words)

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