Relevant bibliographies by topics / Eigenvalue-eigenvector problem

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Eigenvalue-eigenvector problem'

Author: Grafiati
Published: 7 June 2025
Last updated: 16 July 2025

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Journal articles on the topic "Eigenvalue-eigenvector problem"

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Verbitskyi, V. V., and A. G. Huk. "Newton's method for the eigenvalue problem of a symmetric matrix." Researches in Mathematics and Mechanics 25, no. 2(36) (2021): 75–82. http://dx.doi.org/10.18524/2519-206x.2020.2(36).233787.

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Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.
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Jung, Dong-Won, Wooyong Han, U.-Rae Kim, Jungil Lee, and Chaehyun Yu. "Finding normal modes of a loaded string with the method of Lagrange multipliers." Journal of the Korean Physical Society 79, no. 12 (2021): 1079–88. http://dx.doi.org/10.1007/s40042-021-00314-9.

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AbstractWe consider the normal mode problem of a vibrating string loaded with n identical beads of equal spacing, which involves an eigenvalue problem. Unlike the conventional approach to solving this problem by considering the difference equation for the components of the eigenvector, we modify the eigenvalue equation by introducing matrix-valued Lagrange undetermined multipliers, which regularize the secular equation and make the eigenvalue equation non-singular. Then, the eigenvector can be obtained from the regularized eigenvalue equation by multiplying the indeterminate eigenvalue equation by the inverse matrix. We find that the inverse matrix is nothing but the adjugate matrix of the original matrix in the secular determinant up to the determinant of the regularized matrix in the limit that the constraint equation vanishes. The components of the adjugate matrix can be represented in simple factorized forms. Finally, one can directly read off the eigenvector from the adjugate matrix. We expect this new method to be applicable to other eigenvalue problems involving more general forms of the tridiagonal matrices that appear in classical mechanics or quantum physics.
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Cai, Yunfeng, Lei-Hong Zhang, Zhaojun Bai, and Ren-Cang Li. "On an Eigenvector-Dependent Nonlinear Eigenvalue Problem." SIAM Journal on Matrix Analysis and Applications 39, no. 3 (2018): 1360–82. http://dx.doi.org/10.1137/17m115935x.

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Elias, Uri, and Allan Pinkus. "Nonlinear eigenvalue–eigenvector problems for STP matrices." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 6 (2002): 1307–31. http://dx.doi.org/10.1017/s0308210500002122.

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Let Ai, i = 1, …, m, be a set of Ni × Ni−1 strictly totally positive (STP) matrices, with N0 = Nm = N. For a vector x = (x1, …, xN) ∈ RN and arbitrary p > 0, set We consider the eigenvalue-eigenvector problem where p1 … pm−1 = r. We prove an analogue of the classical Gantmacher-Krein theorem for the eigenvalue-eigenvector structure of STP matrices in the case where pi ≥ 1 for each i, plus various extensions thereof.
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Szlachtowska, Ewa, and Dominik Mielczarek. "Eigenvalue problem for the weighted p-Laplacian." Tatra Mountains Mathematical Publications 63, no. 1 (2015): 269–81. http://dx.doi.org/10.1515/tmmp-2015-0037.

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Abstract In this paper we are concerned with the nonlinear eigenvalue problem for the weighted p-Laplacian. The main result of the work is the existence of the eigenpair for the Dirichlet problem provided that the weights are bounded. Furthermore, under this assumption the eigenvector belongs to the Sobolev space W1,p0 (Ω).
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Capdeboscq, Yves. "Homogenization of a neutronic critical diffusion problem with drift." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 3 (2002): 567–94. http://dx.doi.org/10.1017/s0308210500001785.

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In this paper we study the homogenization of an eigenvalue problem for a cooperative system of weakly coupled elliptic partial differential equations, called the neutronic multigroup diffusion model, in a periodic heterogeneous domain. Such a model is used for studying the criticality of nuclear reactor cores. In a recent work in collaboration with Grégoire Allaire, it is proved that, under a symmetry assumption, the first eigenvector of the multigroup system in the periodicity cell controls the oscillatory behaviour of the solutions, whereas the global trend is asymptotically given by a homogenized diffusion eigenvalue problem. It is shown here that when this symmetry condition is not fulfilled, the asymptotic behaviour of the neutron flux, corresponding to the first eigenvector of the multigroup system, is dramatically different. This result enables to consider new types of geometrical configurations in practical nuclear reactor core computations.
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Schwarz, Angelika, and Lars Karlsson. "Scalable eigenvector computation for the non-symmetric eigenvalue problem." Parallel Computing 85 (July 2019): 131–40. http://dx.doi.org/10.1016/j.parco.2019.04.001.

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Laurie, Dirk P. "Solving the inverse eigenvalue problem via the eigenvector matrix." Journal of Computational and Applied Mathematics 35, no. 1-3 (1991): 277–89. http://dx.doi.org/10.1016/0377-0427(91)90214-5.

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HOLZRICHTER, MICHAEL, and SUELY OLIVEIRA. "A GRAPH BASED DAVIDSON ALGORITHM FOR THE GRAPH PARTITIONING PROBLEM." International Journal of Foundations of Computer Science 10, no. 02 (1999): 225–46. http://dx.doi.org/10.1142/s0129054199000162.

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The problem of partitioning a graph such that the number of edges incident to vertices in different partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing fill-in in matrix factorizations and load-balancing for parallel algorithms. Spectral graph partitioning algorithms partition a graph using the eigenvector associated with the second smallest eigenvalue of a matrix called the graph Laplacian. The focus of this paper is the use graph theory to compute this eigenvector more quickly.
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Siswanto, Siswanto. "THE EXISTENCE OF SOLUTION OF GENERALIZED EIGENPROBLEM IN INTERVAL MAX-PLUS ALGEBRA." BAREKENG: Jurnal Ilmu Matematika dan Terapan 17, no. 3 (2023): 1341–46. http://dx.doi.org/10.30598/barekengvol17iss3pp1341-1346.

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An eigenproblem of a matrix is where and . Vector and are eigenvector and eigenvalue, respectively. General form of eigenvalue problem is with , . Interval maks-plus algebra is and equipped with a maximum ( and plus operations. The set of matrices which its component elements of is called matrices over interval max-plus algebra and denoted by . Let , eigenproblem in interval max-plus algebra is with and . Vector and are eigenvector and eigenvalue, respectively. In this research, we will discuss the generalization of the eigenproblem in interval max-plus algebra. Especially about the existence of solution of generalized eigenproblem in interval max-plus algebra. Keywords: interval max-plus algebra, generalized eigenproblem, the existence of the solution.
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Dissertations / Theses on the topic "Eigenvalue-eigenvector problem"

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Kurus, Gulay. "Solution Of Helmholtz Type Equations By Differential Quadarature Method." Master's thesis, METU, 2000. http://etd.lib.metu.edu.tr/upload/2/12605383/index.pdf.

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This thesis presents the Differential Quadrature Method (DQM) for solving Helmholtz, modified Helmholtz and Helmholtz eigenvalue-eigenvector equations. The equations are discretized by using Polynomial-based and Fourier-based differential quadrature technique wich use basically polynomial interpolation for the solution of differential equation.
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Books on the topic "Eigenvalue-eigenvector problem"

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Vlase, Sorin, Marin Marin, and Andreas Öchsner. Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-00991-5.

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Marin, Marin, Andreas Öchsner, and Sorin Vlase. Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer, 2018.

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Marin, Marin, Andreas Öchsner, and Sorin Vlase. Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer, 2019.

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Book chapters on the topic "Eigenvalue-eigenvector problem"

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Vlase, Sorin, Marin Marin, and Andreas Öchsner. "Vectors." In Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00991-5_1.

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Vlase, Sorin, Marin Marin, and Andreas Öchsner. "Matrices." In Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00991-5_2.

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Vlase, Sorin, Marin Marin, and Andreas Öchsner. "Quadratic Forms." In Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00991-5_3.

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Vlase, Sorin, Marin Marin, and Andreas Öchsner. "Rigid Body Mechanics." In Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00991-5_4.

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Vlase, Sorin, Marin Marin, and Andreas Öchsner. "Strain and Stress." In Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00991-5_5.

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Vlase, Sorin, Marin Marin, and Andreas Öchsner. "Modal Analysis." In Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00991-5_6.

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Vlase, Sorin, Marin Marin, and Andreas Öchsner. "Dynamical Systems." In Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00991-5_7.

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Eshel, Gidon. "The Algebraic Operation of SVD." In Spatiotemporal Data Analysis. Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691128917.003.0005.

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Chapter 4 discussed the eigenvalue/eigenvector diagonalization of a matrix. Perhaps the biggest problem for this to be very useful in data analysis is the restriction to square matrices. It has already been emphasized time and again that data matrices, unlike dynamical operators, are rarely square. The algebraic operation of the singular value decomposition (SVD) is the answer. Note the distinction between the data analysis method widely known as SVD and the actual algebraic machinery. The former uses the latter, but is not the latter. This chapter describes the method. Following the introduction to SVD, it provides some examples and applications.
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Ltaief Hatem, Luszczek Piotr, Haidar Azzam, and Dongarra Jack. "Solving the Generalized Symmetric Eigenvalue Problem using Tile Algorithms on Multicore Architectures." In Advances in Parallel Computing. IOS Press, 2012. https://doi.org/10.3233/978-1-61499-041-3-397.

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This paper proposes an efficient implementation of the generalized symmetric eigenvalue problem on multicore architecture. Based on a four-stage approach and tile algorithms, the original problem is first transformed into a standard symmetric eigenvalue problem by computing the Cholesky factorization of the right hand side symmetric definite positive matrix (first stage), and applying the inverse of the freshly computed triangular Cholesky factors to the original dense symmetric matrix of the problem (second stage). Calculating the eigenpairs of the resulting problem is then equivalent to the eigenpairs of the original problem. The computation proceeds by reducing the updated dense symmetric matrix to symmetric band form (third stage). The band structure is further reduced by applying a bulge chasing procedure, which annihilates the extra off-diagonal entries using orthogonal transformations (fourth stage). More details on the third and fourth stage can be found in Haidar et al. [Accepted at SC'11, November 2011]. The eigenvalues are then calculated from the tridiagonal form using the standard LAPACK QR algorithm (i.e., DTSEQR routine), while the complex and challenging eigenvector computations will be addressed in a companion paper. The tasks from the various stages can concurrently run in an out-of-order fashion. The data dependencies are cautiously tracked by the dynamic runtime system environment QUARK, which ensures the dependencies are not violated for numerical correctness purposes. The obtained tile four-stage generalized symmetric eigenvalue solver significantly outperforms the state-of-the-art numerical libraries (up to 21-fold speed up against multithreaded LAPACK with optimized multithreaded MKL BLAS and up to 4-fold speed up against the corresponding routine from the commercial numerical software Intel MKL) on four sockets twelve cores AMD system with a 24000×24000 matrix size.
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Vía, Javier, Ignacio Santamaría, and Jesús Ibáñez. "Blind Channel Estimation in Space-Time Block Coded Systems." In Handbook on Advancements in Smart Antenna Technologies for Wireless Networks. IGI Global, 2009. http://dx.doi.org/10.4018/978-1-59904-988-5.ch008.

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This chapter analyzes the problem of blind channel estimation under Space-Time Block Coded transmissions. In particular, a new blind channel estimation technique for a general class of space-time block codes is proposed. The method is solely based on the second-order statistics of the observations, and its computational complexity reduces to the extraction of the main eigenvector of a generalized eigenvalue problem. Additionally, the identifiability conditions associated to the blind channel estimation problem are analyzed, which is exploited to propose a new transmission technique based on the idea of code diversity or combination of different codes. This technique resolves the ambiguities in most of the practical cases, and it can be reduced to a non-redundant precoding consisting in a single set of rotations or permutations of the transmit antennas. Finally, the performance of the proposed techniques is illustrated by means of several simulation examples.
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Conference papers on the topic "Eigenvalue-eigenvector problem"

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Feeny, B. F., and U. Farooq. "A State-Variable Decomposition Method for Estimating Modal Parameters." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35651.

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A nonsymmetric, generalized eigenvalue problem is constructed from state-variable ensembles. The data-based eigenvalue problem is related to the state-variable formulation of linear multi-degree-of-freedom systems. The inverse-transpose of the eigenvector matrix from this eigenvalue problem converges to the state-variable modal eigenvectors, and the eigenvalues lead to estimates of frequencies and modal damping. The interpretation holds whether damping is modal or nonmodal, and without the need of input data. The method is illustrated on an eight degree-of-freedom mass-spring-dashpot example.
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Bravo, Ignacio, Pedro Jimenez, Manuel Mazo, Jose Luis Lazaro, and Alfredo Gardel. "Implementation in Fpgas of Jacobi Method to Solve the Eigenvalue and Eigenvector Problem." In 2006 International Conference on Field Programmable Logic and Applications. IEEE, 2006. http://dx.doi.org/10.1109/fpl.2006.311301.

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Ekici, Ekrem, and Matthew P. Juniper. "Adjoint Based Shape Optimization for Thermoacoustic Stability of Combustors Using Free Form Deformation." In ASME Turbo Expo 2024: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2024. http://dx.doi.org/10.1115/gt2024-125711.

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Abstract We use the thermoacoustic Helmholtz equation to model thermoacoustic oscillations as an eigenvalue problem. We solve this with a Finite Element method. We parameterize the geometry of an annular combustor geometry using Free Form Deformation (FFD). We then use the FFD geometry, define the system parameters and impose the acoustic boundary conditions to calculate the eigenvalue and eigenvector of the problem using a Helmholtz solver. We then use adjoint methods to calculate the shape derivatives of the unstable eigenvalue with respect to the FFD control points. According to these gradients, we propose modifications to the control points that reduce the growth rate. We first demonstrate the application of this approach on the Rijke tube. Then we extend the method to a simulation of a laboratory combustor and lower the growth rate of the unstable circumferential mode. These findings show how this method could be used to reduce combustion instability in industrial annular combustors through geometric modifications.
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Zhou, Gaoxin, and Zhi Gang. "Refined Algorithm for Solving High-Order Harmonics of Multi Group Neutron Diffusion Equation." In 2017 25th International Conference on Nuclear Engineering. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/icone25-66098.

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In recent years, high order harmonic (or eigenvector) of neutron diffusion equation has been widely used in on-line monitoring system of reactor power. There are two kinds of calculation method to solve the equation: corrected power iteration method and Krylov subspace methods. Fu Li used the corrected power iteration method. When solving for the ith harmonic, it tries to eliminate the influence of the front harmonics using the orthogonality of the harmonic function. But its convergence speed depends on the occupation ratio. When the dominant ratios equal to 1 or close to 1, convergence speed of fixed source iteration method is slow or convergence can’t be achieved. Another method is the Krylov subspace method, the main idea of this method is to project the eigenvalue and eigenvector of large-scale matrix to a small one. Then we can solve the small matrix eigenvalue and eigenvector to get the large ones. In recent years, the restart Arnoldi method emerged as a development of Krylov subspace method. The method uses continuous reboot Arnoldi decomposition, limiting expanding subspace, and the orthogonality of the subspace is guaranteed using orthogonalization method. This paper studied the refined algorithms, a method based on the Krylov subspace method of solving eigenvalue problem for large sparse matrix of neutron diffusion equation. Two improvements have been made for a restarted Arnoldi method. One is that using an ingenious linear combination of the refined Ritz vector forms an initial vector and then generates a new Krylov subspace. Another is that retaining the refined Ritz vector in the new subspace, called, augmented Krylov subspace. This way retains useful information and makes the resulting algorithm converge faster. Several numerical examples are the new algorithm with the implicitly restart Arnoldi algorithm (IRA) and the implicitly restarted refined Arnoldi algorithm (IRRA). Numerical results confirm efficiency of the new algorithm.
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Xu, Zhiqiang, Xin Cao, and Xin Gao. "Convergence Analysis of Gradient Descent for Eigenvector Computation." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/407.

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We present a novel, simple and systematic convergence analysis of gradient descent for eigenvector computation. As a popular, practical, and provable approach to numerous machine learning problems, gradient descent has found successful applications to eigenvector computation as well. However, surprisingly, it lacks a thorough theoretical analysis for the underlying geodesically non-convex problem. In this work, the convergence of the gradient descent solver for the leading eigenvector computation is shown to be at a global rate O(min{ (lambda_1/Delta_p)^2 log(1/epsilon), 1/epsilon }), where Delta_p=lambda_p-lambda_p+1>0 represents the generalized positive eigengap and always exists without loss of generality with lambda_i being the i-th largest eigenvalue of the given real symmetric matrix and p being the multiplicity of lambda_1. The rate is linear at (lambda_1/Delta_p)^2 log(1/epsilon) if (lambda_1/Delta_p)^2=O(1), otherwise sub-linear at O(1/epsilon). We also show that the convergence only logarithmically instead of quadratically depends on the initial iterate. Particularly, this is the first time the linear convergence for the case that the conventionally considered eigengap Delta_1= lambda_1 - lambda_2=0 but the generalized eigengap Delta_p satisfies (lambda_1/Delta_p)^2=O(1), as well as the logarithmic dependence on the initial iterate are established for the gradient descent solver. We are also the first to leverage for analysis the log principal angle between the iterate and the space of globally optimal solutions. Theoretical properties are verified in experiments.
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Weng, Lin, Zengliang Gao, and Xiaogui Wang. "Determination of the Stress Intensity Factors for Axisymmetric Cylindrical Interface Crack by an Eigenvalue Method." In ASME 2009 Pressure Vessels and Piping Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/pvp2009-78014.

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An eigenvalue method was proposed to study the stress intensity factors associated with the oscillating stress singularity for the axisymmetric cylindrical interface crack of the fiber/matrix composites. The fiber is a transversely isotropic material and the matrix is isotropic. Based on the fundamental equations of the spacial axisymmetric problem and the assumption of first-order approximation of the singular stress field, the discrete characteristic equation was derived using the displacement functions in the form of separated variables and the technique of meshless method. The eigenvalue is relative to the order of stress singularity, and the associated eigenvector is with respect to the displacement angular variations. The stress angular variations were derived by introducing the displacement angular variations into the constitutive relations. A finite element fiber/matrix model was used to verify the validation of the proposed eigenvalue method. The numerical results of the order of stress singularity and normalized stress angular variations are in good agreement with those obtained by the eigenvalue method. Based on the order of stress singularity and stress angular variations obtained by the eigenvalue method, as well as the numerical singular stress fields obtained by the finite element method (FEM), the stress intensity factors were determined successfully with the linear extropolation method.
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Wu, T. Y., and K. W. Wang. "Vibration Isolation Control Via Simultaneous Left and Right Eigenvector Assignment." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84824.

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The objective of this research is to investigate the feasibility of utilizing the simultaneous left and right eigenvector assignment concept for vibration isolation feedback control design. The purpose of the right eigenvector assignment method is to alter the closed-loop system modes such that the modal components corresponding to the concerned regions (isolation end of an isolator) have relatively small amplitude. Correspondently, the design goal of left eigenvector assignment is to alter the left eigenvectors of the closed-loop system so that they are as closely orthogonal to the system’s forcing vectors as possible. With this approach, one can achieve both disturbance rejection and modal confinement concurrently for the purpose of vibration isolation. In this research, a new formulation is developed so that the desired left eigenvectors of this integrated system are selected through solving a generalized eigenvalue problem, where the orthogonality indices between the forcing vector and the left eigenvectors are minimized. The components of right eigenvectors corresponding to the concerned regions are minimized concurrently. To realistically implement the algorithm, an integrated closed-loop system with state estimator is developed. Numerical simulations are performed to evaluate the effectiveness of the proposed method on concurrent disturbance rejection and modal confinement for a isolator rod design. Frequency responses of the isolator in the selected frequency range are illustrated. It is shown that with the simultaneous left-right eigenvector assignment technique, both disturbance rejection and modal confinement can be achieved, and thus the vibration amplitude in the isolated regions can be suppressed significantly.
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Ge, Q. J., and B. Ravani. "Computation of Spatial Displacements From Redundant Geometric Features." In ASME 1992 Design Technical Conferences. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/detc1992-0156.

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Abstract This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1992). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.
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Shakarji, Craig M., and Vijay Srinivasan. "Fitting Weighted Total Least-Squares Planes and Parallel Planes to Support Tolerancing Standards." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70398.

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We present elegant algorithms for fitting a plane, two parallel planes (corresponding to a slot or a slab) or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3×3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we demonstrate that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.
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Dan, Byung Ju, and Yong Je Choi. "Locus of the Axis of Vibration for a Vibrating System With Variable Location of a Mass Center." In ASME 2000 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2000. http://dx.doi.org/10.1115/imece2000-2018.

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Abstract A typical approach to a linear vibration analysis of an elastically supported single rigid body is to rearrange a dynamic model into a corresponding eigenvalue problem. From the geometrical point of view, the eigenvectors in the planar vibration analysis can be interpreted as pure rotations about the vibration center or pure translations. In a three dimensional space, they represent repetitive twisting motions about the axes of vibrations. By taking a geometrical approach to the vibration analysis, the vibration mode shapes may be better understood. In this paper, the influence of variable location of a mass center on the locations of the axes of vibrations and the natural frequencies are investigated by means of the locus of the axis of vibration expressed in analytical form, which represents the geometrical locus of the eigenvector. A numerical example is used to clearly illustrate the vibration phenomena of an optical pick-up used in an information storage device.
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