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Journal articles on the topic 'Generalized eigenvalue algorithm'

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Published: 4 June 2021
Last updated: 1 February 2022

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1

Kangal, Fatih, and Emre Mengi. "Nonsmooth algorithms for minimizing the largest eigenvalue with applications to inner numerical radius." IMA Journal of Numerical Analysis 40, no. 4 (November 13, 2019): 2342–76. http://dx.doi.org/10.1093/imanum/drz041.

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Abstract Nonsmoothness at optimal points is a common phenomenon in many eigenvalue optimization problems. We consider two recent algorithms to minimize the largest eigenvalue of a Hermitian matrix dependent on one parameter, both proven to be globally convergent unaffected by nonsmoothness. One of these algorithms models the eigenvalue function with a piece-wise quadratic function and is effective in dealing with nonconvex problems. The other algorithm projects the Hermitian matrix into subspaces formed of eigenvectors and is effective in dealing with large-scale problems. We generalize the latter slightly to cope with nonsmoothness. For both algorithms we analyze the rate of convergence in the nonsmooth setting, when the largest eigenvalue is multiple at the minimizer and zero is strictly in the interior of the generalized Clarke derivative, and prove that both algorithms converge rapidly. The algorithms are applied to, and the deduced results are illustrated on the computation of the inner numerical radius, the modulus of the point on the boundary of the field of values closest to the origin, which carries significance for instance for the numerical solution of a symmetric definite generalized eigenvalue problem and the iterative solution of a saddle point linear system.
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2

Nedamani, F. Abbasi, A. H. Refahi Sheikhani, and H. Saberi Najafi. "A New Algorithm for Solving Large-Scale Generalized Eigenvalue Problem Based on Projection Methods." Mathematical Problems in Engineering 2020 (December 14, 2020): 1–10. http://dx.doi.org/10.1155/2020/8895856.

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In this paper, we consider four methods for determining certain eigenvalues and corresponding eigenvectors of large-scale generalized eigenvalue problems which are located in a certain region. In these methods, a small pencil that contains only the desired eigenvalue is derived using moments that have obtained via numerical integration. Our purpose is to improve the numerical stability of the moment-based method and compare its stability with three other methods. Numerical examples show that the block version of the moment-based (SS) method with the Rayleigh–Ritz procedure has higher numerical stability than respect to other methods.
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3

Zheng, Wenming. "Class-Incremental Generalized Discriminant Analysis." Neural Computation 18, no. 4 (April 1, 2006): 979–1006. http://dx.doi.org/10.1162/neco.2006.18.4.979.

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Generalized discriminant analysis (GDA) is the nonlinear extension of the classical linear discriminant analysis (LDA) via the kernel trick. Mathematically, GDA aims to solve a generalized eigenequation problem, which is always implemented by the use of singular value decomposition (SVD) in the previously proposed GDA algorithms. A major drawback of SVD, however, is the difficulty of designing an incremental solution for the eigenvalue problem. Moreover, there are still numerical problems of computing the eigenvalue problem of large matrices. In this article, we propose another algorithm for solving GDA as for the case of small sample size problem, which applies QR decomposition rather than SVD. A major contribution of the proposed algorithm is that it can incrementally update the discriminant vectors when new classes are inserted into the training set. The other major contribution of this article is the presentation of the modified kernel Gram-Schmidt (MKGS) orthogonalization algorithm for implementing the QR decomposition in the feature space, which is more numerically stable than the kernel Gram-Schmidt (KGS) algorithm. We conduct experiments on both simulated and real data to demonstrate the better performance of the proposed methods.
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4

Oustry, François. "A Superlinear Algorithm to Solve Generalized Eigenvalue Problems." IFAC Proceedings Volumes 30, no. 27 (October 1997): 291–95. http://dx.doi.org/10.1016/s1474-6670(17)41197-9.

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5

Dai, Hua. "An algorithm for symmetric generalized inverse eigenvalue problems." Linear Algebra and its Applications 296, no. 1-3 (July 1999): 79–98. http://dx.doi.org/10.1016/s0024-3795(99)00109-3.

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6

Zheng, Wenming, Li Zhao, and Cairong Zou. "A Modified Algorithm for Generalized Discriminant Analysis." Neural Computation 16, no. 6 (June 1, 2004): 1283–97. http://dx.doi.org/10.1162/089976604773717612.

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Generalized discriminant analysis (GDA) is an extension of the classical linear discriminant analysis (LDA) from linear domain to a nonlinear domain via the kernel trick. However, in the previous algorithm of GDA, the solutions may suffer from the degenerate eigenvalue problem (i.e., several eigenvectors with the same eigenvalue), which makes them not optimal in terms of the discriminant ability. In this letter, we propose a modified algorithm for GDA (MGDA) to solve this problem. The MGDA method aims to remove the degeneracy of GDA and find the optimal discriminant solutions, which maximize the between-class scatter in the subspace spanned by the degenerate eigenvectors of GDA. Theoretical analysis and experimental results on the ORL face database show that the MGDA method achieves better performance than the GDA method.
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7

Rajakumar, C., and C. R. Rogers. "The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem." International Journal for Numerical Methods in Engineering 32, no. 5 (October 5, 1991): 1009–26. http://dx.doi.org/10.1002/nme.1620320506.

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8

Aishima, Kensuke. "A quadratically convergent algorithm for inverse generalized eigenvalue problems." Journal of Computational and Applied Mathematics 367 (March 2020): 112485. http://dx.doi.org/10.1016/j.cam.2019.112485.

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9

Li, Kuiyuan, Tien-Yien Li, and Zhonggang Zeng. "An algorithm for the generalized symmetric tridiagonal eigenvalue problem." Numerical Algorithms 8, no. 2 (September 1994): 269–91. http://dx.doi.org/10.1007/bf02142694.

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10

Spiridonov, Alexander O., Anna I. Repina, Ilya V. Ketov, Sergey I. Solov’ev, and Evgenii M. Karchevskii. "Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes." Axioms 10, no. 3 (August 11, 2021): 184. http://dx.doi.org/10.3390/axioms10030184.

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The paper investigates an algorithm for the numerical solution of a parametric eigenvalue problem for the Helmholtz equation on the plane specially tailored for the accurate mathematical modeling of lasing modes of microring lasers. The original problem is reduced to a nonlinear eigenvalue problem for a system of Muller boundary integral equations. For the numerical solution of the obtained problem, we use a trigonometric Galerkin method, prove its convergence, and derive error estimates in the eigenvalue and eigenfunction approximation. Previous numerical experiments have shown that the method converges exponentially. In the current paper, we prove that if the generalized eigenfunctions are analytic, then the approximate eigenvalues and eigenfunctions exponentially converge to the exact ones as the number of basis functions increases. To demonstrate the practical effectiveness of the algorithm, we find geometrical characteristics of microring lasers that provide a significant increase in the directivity of lasing emission, while maintaining low lasing thresholds.
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11

Wang, Hui, and Zhi Bin Li. "An Inverse Problem of Eigenvalue for Generalized Anti-Tridiagonal Matrices." Advanced Materials Research 424-425 (January 2012): 377–80. http://dx.doi.org/10.4028/www.scientific.net/amr.424-425.377.

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An inverse problem of eigenvalue for generalized Anti-Tridiagonal Matrices is discussed on the base of some inverse problems of Eigenvalue for Anti-Tridiagonal Matrices. The algorithm and uniqueness theorem of the solution of the problem are given, and some numerical example is provided
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12

Yin, Guojian. "A harmonic FEAST algorithm for non-Hermitian generalized eigenvalue problems." Linear Algebra and its Applications 578 (October 2019): 75–94. http://dx.doi.org/10.1016/j.laa.2019.04.036.

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13

Sundar, S., and B. K. Bhagavan. "Generalized eigenvalue problems: Lanczos algorithm with a recursive partitioning method." Computers & Mathematics with Applications 39, no. 7-8 (April 2000): 211–24. http://dx.doi.org/10.1016/s0898-1221(00)00077-8.

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14

Mathew, G., and V. U. Reddy. "A quasi-Newton adaptive algorithm for generalized symmetric eigenvalue problem." IEEE Transactions on Signal Processing 44, no. 10 (1996): 2413–22. http://dx.doi.org/10.1109/78.539027.

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15

Yin, Guojian, Raymond H. Chan, and Man-Chung Yeung. "A FEAST algorithm with oblique projection for generalized eigenvalue problems." Numerical Linear Algebra with Applications 24, no. 4 (March 5, 2017): e2092. http://dx.doi.org/10.1002/nla.2092.

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16

Bello, Lenys, William La Cruz, and Marcos Raydan. "Residual algorithm for large-scale positive definite generalized eigenvalue problems." Computational Optimization and Applications 46, no. 2 (April 24, 2009): 217–27. http://dx.doi.org/10.1007/s10589-009-9250-9.

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17

Attallah, Samir, and Karim Abed-Meraim. "A Fast Adaptive Algorithm for the Generalized Symmetric Eigenvalue Problem." IEEE Signal Processing Letters 15 (2008): 797–800. http://dx.doi.org/10.1109/lsp.2008.2006346.

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18

Sagitov, M. S. "Finite splitting algorithm for the j-symmetric generalized eigenvalue problem." USSR Computational Mathematics and Mathematical Physics 30, no. 6 (January 1990): 119–26. http://dx.doi.org/10.1016/0041-5553(90)90119-d.

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19

Gupta, Aashu, Dr Vijay Lamba, and Er Munish Verma. "Design of IIR Filter using Remez Algorithm." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 3, no. 1 (August 1, 2012): 117–20. http://dx.doi.org/10.24297/ijct.v3i1b.2751.

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In this paper, we present a numerical method for the equiripple approximation of Impulse Infinite Response digital filters. The proposed method is based on the formulation of a generalized eigenvalue problem by using Rational Remez Exchange algorithm. In this paper, conventional Remez algorithm is modified to get the ratio of weights in the different bands exactly. In Rational Remez, squared magnitude response of the IIR filter is approximated in the Chebyshev sense by solving for an eigenvalue problem, in which real maximum eigenvalue is chosen and corresponding to that eigenvectors are found, and from that optimal filter coefficients are obtained through few iterations with controlling the ratio of ripples. The design algorithm is computationally efficient because it not only retains the speed inherent in the Remez exchange algorithm but also simplifies the interpolation step.
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20

Karpfinger, Florian, Henri-Pierre Valero, Boris Gurevich, Andrey Bakulin, and Bikash Sinha. "Spectral-method algorithm for modeling dispersion of acoustic modes in elastic cylindrical structures." GEOPHYSICS 75, no. 3 (May 2010): H19—H27. http://dx.doi.org/10.1190/1.3380590.

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A new spectral-method algorithm can be used to study wave propagation in cylindrically layered fluid and elastic structures. The cylindrical structure is discretized with Chebyshev points in the radial direction, whereas differentiation matrices are used to approximate the differential operators. We express the problem of determining modal dispersions as a generalized eigenvalue problem that can be solved readily for all eigenvalues corresponding to various axial wavenumbers. Modal dispersions of guided modes can then be expressed in terms of axial wavenumbers as a function of frequency. The associated eigenvectors are related to the displacement potentials that can be used to calcu-late radial distributions of modal amplitudes as well as stress components at a given frequency. The workflow includes input parameters and the construction of differentiation matrices and boundary conditions that yield the generalized eigenvalue problem. Results from this algorithm for a fluid-filled borehole surrounded by an elastic formation agree very well with those from a root-finding search routine. Computational efficiency of the algorithm has been demonstrated on a four-layer completion model used in a hydrocarbon-producing well. Even though the algorithm is numerically unstable at very low frequencies, it produces reliable and accurate results for multilayered cylindrical structures at moderate frequencies that are of interest in estimating formation properties using modal dispersions.
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21

Wang, Zhen, and Dong Mei Li. "Multiple-Instance Classification via Generalized Eigenvalue Proximal SVM." Advanced Materials Research 143-144 (October 2010): 1235–39. http://dx.doi.org/10.4028/www.scientific.net/amr.143-144.1235.

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The multiple-instance classification problem is formulated using a linear or nonlinear kernel as the minimization of a linear function in a finite dimensional real space subject to linear and bilinear constraints by SVM-based methods. This paper presents a new multiple-instance classifier that determines two nonparallel planes by solving generalized eigenvalue proximal SVM. Our method converges in a few iterations to a local solution. Computational results on a number of datasets indicate that the proposed algorithm is competitive with the other SVM-based methods in multiple-instance classification.
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22

Kalantzis, Vassilis. "A Domain Decomposition Rayleigh--Ritz Algorithm for Symmetric Generalized Eigenvalue Problems." SIAM Journal on Scientific Computing 42, no. 6 (January 2020): C410—C435. http://dx.doi.org/10.1137/19m1280004.

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23

Luo, Xin-long, Jia-ru Lin, and Wei-ling Wu. "A Prediction-Correction Dynamic Method for Large-Scale Generalized Eigenvalue Problems." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/845459.

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This paper gives a new prediction-correction method based on the dynamical system of differential-algebraic equations for the smallest generalized eigenvalue problem. First, the smallest generalized eigenvalue problem is converted into an equivalent-constrained optimization problem. Second, according to the Karush-Kuhn-Tucker conditions of this special equality-constrained problem, a special continuous dynamical system of differential-algebraic equations is obtained. Third, based on the implicit Euler method and an analogous trust-region technique, a prediction-correction method is constructed to follow this system of differential-algebraic equations to compute its steady-state solution. Consequently, the smallest generalized eigenvalue of the original problem is obtained. The local superlinear convergence property for this new algorithm is also established. Finally, in comparison with other methods, some promising numerical experiments are presented.
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24

Mesloub, Ammar. "Generalized Givens Rotations Applied to Complex Joint Eigenvalue Decomposition." ENP Engineering Science Journal 1, no. 1 (July 22, 2021): 58–62. http://dx.doi.org/10.53907/enpesj.v1i1.49.

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This paper shows the different ways of using generalized Givens rotations in complex joint eigenvaluedecomposition (JEVD) problem. It presents the different schemes of generalized Givens rotation, justifies the introducedapproximations and focuses on the process of extending an algorithm developed for real JEVD to the complex JEVD.Several Joint Diagonalization problem use generalized Givens rotations to achieve the solution, many algorithmsdeveloped in the real case exist in the literature and are not generalized to the complex case. Hence, we show herein asimple and not trivial way to get the complex case from the real one. Simulation results are provided to highlight theeffectiveness and behaviour of the proposed techniques for different scenarios.
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25

Pillai, V. K., and H. D. Nelson. "A New Algorithm for Limit Cycle Analysis of Nonlinear Control Systems." Journal of Dynamic Systems, Measurement, and Control 110, no. 3 (September 1, 1988): 272–77. http://dx.doi.org/10.1115/1.3152681.

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A numerical method is presented for the limit cycle analysis of multiloop nonlinear control systems with multiple nonlinearities. Describing functions are used to model the first harmonic gains of the nonlinearities. Existence of a limit cycle is sought by driving the least damped eigenvalues to the imaginary axis. The evolution of the limit cycle is studied next as a function of a critical system-parameter. It is shown that by defining a suitable error function it is possible to use both eigenvalue as well as the eigenvector sensitivities to formulate a generalized Newton-Raphson method to solve simultaneously for the updates of state variable amplitudes in a minimum norm sense. Several case studies have been presented and the development of a numerical procedure to test the stability of the limit cycle has also been reported.
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26

Chandrasekaran, S. "An Efficient and Stable Algorithm for the Symmetric-Definite Generalized Eigenvalue Problem." SIAM Journal on Matrix Analysis and Applications 21, no. 4 (January 2000): 1202–28. http://dx.doi.org/10.1137/s0895479897316308.

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27

Saberi Najafi, H., and A. Refahi. "A new restarting method in the Lanczos algorithm for generalized eigenvalue problem." Applied Mathematics and Computation 184, no. 2 (January 2007): 421–28. http://dx.doi.org/10.1016/j.amc.2006.06.079.

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28

Kalantzis, Vassilis. "A spectral Newton-Schur algorithm for the solution of symmetric generalized eigenvalue problems." ETNA - Electronic Transactions on Numerical Analysis 52 (2020): 132–53. http://dx.doi.org/10.1553/etna_vol52s132.

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29

Ferng, William R., Kun-Yi Lin, and Wen-Wei Lin. "A novel nonsymmetric K−-Lanczos algorithm for the generalized nonsymmetric K−-eigenvalue problems." Linear Algebra and its Applications 252, no. 1-3 (February 1997): 81–105. http://dx.doi.org/10.1016/s0024-3795(96)00670-2.

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30

Natarajan, Ramesh. "A parallel algorithm for the generalized symmetric eigenvalue problem on a hybrid multiprocessor." Parallel Computing 14, no. 2 (June 1990): 129–50. http://dx.doi.org/10.1016/0167-8191(90)90103-g.

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31

H.Marghny, M., Rasha M. Abd El-Aziz, and Ahmed I. Taloba. "Differential Search Algorithm-based Parametric Optimization of Fuzzy Generalized Eigenvalue Proximal Support Vector Machine." International Journal of Computer Applications 108, no. 19 (December 18, 2014): 38–46. http://dx.doi.org/10.5120/19023-0540.

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32

Hwang, T., and I. D. Parsons. "A multigrid method for the generalized symmetric eigenvalue problem: Part I—algorithm and implementation." International Journal for Numerical Methods in Engineering 35, no. 8 (November 15, 1992): 1663–76. http://dx.doi.org/10.1002/nme.1620350807.

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33

Dai, Hua, Zhong-Zhi Bai, and Ying Wei. "On the Solvability Condition and Numerical Algorithm for the Parameterized Generalized Inverse Eigenvalue Problem." SIAM Journal on Matrix Analysis and Applications 36, no. 2 (January 2015): 707–26. http://dx.doi.org/10.1137/140972494.

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34

Bosner, Nela. "Parallel reduction of four matrices to condensed form for a generalized matrix eigenvalue algorithm." Numerical Algorithms 86, no. 1 (March 2, 2020): 153–78. http://dx.doi.org/10.1007/s11075-020-00883-z.

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35

Chopra, Satinder, and Kurt J. Marfurt. "Multispectral, multiazimuth, and multioffset coherence attribute applications." Interpretation 7, no. 2 (May 1, 2019): SC21—SC32. http://dx.doi.org/10.1190/int-2018-0090.1.

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The coherence attribute computation is typically carried out as a poststack application on 3D prestack migrated seismic data volumes. However, since its inception, interpreters have applied coherence to band-pass-filtered data, azimuthally limited stacks, and offset-limited stacks to enhance discontinuities seen at specific frequencies, azimuths, and offsets. The limitation of this approach is the multiplicity of coherence volumes. Of the various coherence algorithms that have evolved over the past 25 years, the energy ratio coherence computation stands apart from the others, being more sensitive to the seismic waveform changes rather than changes in their amplitude. The energy ratio algorithm is based on the crosscorrelation of five or more adjacent traces to form a symmetric covariance matrix that can then be decomposed into eigenvalues and eigenvectors. The first eigenvector represents a vertically variable, laterally consistent pattern that best represents the data in the analysis window. The first eigenvalue represents the energy of the data represented by this pattern. Coherence is then defined as the ratio of the energy represented by the first eigenvalue to the sum of the energy of the original data. An early generalization of this algorithm was to compute the sum of two covariance matrices, one from the original data and the other from the 90° phase rotated data, thereby eliminating artifacts about low-amplitude zero crossings. More recently, this concept has been further generalized by computing a sum of covariance matrices of traces represented by multiple spectral components, by their azimuthally limited stacks, and by their offset-limited stacks. These more recently developed algorithms capture many of the benefits of discontinuities seen at specific frequencies, azimuths, and offsets, but they present the interpreter with a single volume. We compare the results of multispectral, multiazimuth, and multioffset coherence volumes with the traditional coherence computation, and we find that these newer coherence computation procedures produce superior results.
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36

Gadzhiev, Magomed, Misrikhan Misrikhanov, Vladimir Ryabchenko, and Nikita Vasilenko. "Randomized algorithm of invariant zeros evaluation of electrical power system, defined in descriptor form." E3S Web of Conferences 58 (2018): 01004. http://dx.doi.org/10.1051/e3sconf/20185801004.

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A randomized algorithm for computing the invariant zeros of the electric energy system as a dynamical system with many inputs and many outputs (MIMO system), specified in the descriptor form, is proposed. Definitions of invariant zeros are carried out by randomizing the original MIMO system and it reduces to a generalized eigenvalue problem for a numerical matrix. The application of the algorithm is illustrated by the example of calculating the invariant zeros of the linear model of the United Power System.
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37

Li, Fan-Liang. "Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix." Open Mathematics 18, no. 1 (June 18, 2020): 603–15. http://dx.doi.org/10.1515/math-2020-0020.

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Abstract Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.
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38

Ahn, Jong-Hoon, and Jong-Hoon Oh. "A Constrained EM Algorithm for Principal Component Analysis." Neural Computation 15, no. 1 (January 1, 2003): 57–65. http://dx.doi.org/10.1162/089976603321043694.

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We propose a constrained EM algorithm for principal component analysis (PCA) using a coupled probability model derived from single-standard factor analysis models with isotropic noise structure. The single probabilistic PCA, especially for the case where there is no noise, can find only a vector set that is a linear superposition of principal components and requires postprocessing, such as diagonalization of symmetric matrices. By contrast, the proposed algorithm finds the actual principal components, which are sorted in descending order of eigenvalue size and require no additional calculation or postprocessing. The method is easily applied to kernel PCA. It is also shown that the new EM algorithm is derived from a generalized least-squares formulation.
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39

Fathi Vajargah, Behrouz, and Farshid Mehrdoust. "Partitioning Inverse Monte Carlo Iterative Algorithm for Finding the Three Smallest Eigenpairs of Generalized Eigenvalue Problem." Advances in Numerical Analysis 2011 (April 11, 2011): 1–9. http://dx.doi.org/10.1155/2011/826376.

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A new Monte Carlo approach for evaluating the generalized eigenpair of real symmetric matrices will be proposed. Algorithm for the three smallest eigenpairs based on the partitioning inverse Monte Carlo iterative (IMCI) method will be considered.
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40

Ito, Shinji, and Kazuo Murota. "An Algorithm for the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils by Minimal Perturbation Approach." SIAM Journal on Matrix Analysis and Applications 37, no. 1 (January 2016): 409–19. http://dx.doi.org/10.1137/14099231x.

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41

Maeda, Kazuki, and Satoshi Tsujimoto. "A generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system." Journal of Computational and Applied Mathematics 300 (July 2016): 134–54. http://dx.doi.org/10.1016/j.cam.2015.12.032.

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42

Cao, Jianzhi, Peiguang Wang, Rong Yuan, and Yingying Mei. "Bogdanov–Takens Bifurcation of a Class of Delayed Reaction–Diffusion System." International Journal of Bifurcation and Chaos 25, no. 06 (June 15, 2015): 1550082. http://dx.doi.org/10.1142/s0218127415500820.

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In this paper, a class of reaction–diffusion system with Neumann boundary condition is considered. By analyzing the generalized eigenvector associated with zero eigenvalue, an equivalent condition for the determination of Bogdonov–Takens (B–T) singularity is obtained. Next, by using center manifold theorem and normal form method, we have a two-dimension ordinary differential system on its center manifold. Finally, two examples show that the given algorithm is effective.
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43

Teng, Zhongming, and Xiaowei Zhang. "A Jacobi–Davidson Method for Large Scale Canonical Correlation Analysis." Algorithms 13, no. 9 (September 12, 2020): 229. http://dx.doi.org/10.3390/a13090229.

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In the large scale canonical correlation analysis arising from multi-view learning applications, one needs to compute canonical weight vectors corresponding to a few of largest canonical correlations. For such a task, we propose a Jacobi–Davidson type algorithm to calculate canonical weight vectors by transforming it into the so-called canonical correlation generalized eigenvalue problem. Convergence results are established and reveal the accuracy of the approximate canonical weight vectors. Numerical examples are presented to support the effectiveness of the proposed method.
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44

Do, Minh-Hieu, Patrick Ciarlet, and François Madiot. "ADAPTIVE SOLUTION OF THE NEUTRON DIFFUSION EQUATION WITH HETEROGENEOUS COEFFICIENTS USING THE MIXED FINITE ELEMENT METHOD ON STRUCTURED MESHES." EPJ Web of Conferences 247 (2021): 02002. http://dx.doi.org/10.1051/epjconf/202124702002.

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The neutron transport equation can be used to model the physics of the nuclear reactor core. Its solution depends on several variables and requires a lot of high precision computations. One can simplify this model to obtain the SPN equation for a generalized eigenvalue problem. In order to solve this eigenvalue problem, we usually use the inverse power iteration by solving a source problem at each iteration. Classically, this problem can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. In this article, we focus on the steady-state diffusion equation with heterogeneous coefficients discretized on Cartesian meshes. In this situation, it is expected that the solution has low regularity. Therefore, it is necessary to refine at the singular regions to get better accuracy. The Adaptive Mesh Refinement (AMR) is one of the most effective ways to do that and to improve the computational time. The main ingredient for the refinement techniques is the use of a posteriori error estimates, which gives a rigorous upper bound of the error between the exact and numerical solution. This indicator allows to refine the mesh in the regions where the error is large. In this work, some mesh refinement strategies are proposed on the Cartesian mesh for the source problem. Moreover, we numerically investigate an algorithm which combines the AMR process with the inverse power iteration to handle the generalized eigenvalue problem.
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45

Ma, Zheng-Dong, Noboru Kikuchi, Hsien-Chie Cheng, and Ichiro Hagiwara. "Topological Optimization Technique for Free Vibration Problems." Journal of Applied Mechanics 62, no. 1 (March 1, 1995): 200–207. http://dx.doi.org/10.1115/1.2895903.

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A topological optimization technique using the conception of OMD (Optimal Material Distribution) is presented for free vibration problems of a structure. A new objective function corresponding to multieigenvalue optimization is suggested for improving the solution of the eigenvalue optimization problem. An improved optimization algorithm is then applied to solve these problems, which is derived by the authors using a new convex generalized-linearization approach via a shift parameter which corresponds to the Lagrange multiplier and the use of the dual method. Finally, three example applications are given to substantiate the feasibility of the approaches presented in this paper.
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46

Deift, P. A., and L. C. Li. "Generalized affine lie algebras and the solution of a class of flows associated with the QR eigenvalue algorithm." Communications on Pure and Applied Mathematics 42, no. 7 (October 1989): 963–91. http://dx.doi.org/10.1002/cpa.3160420704.

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47

Savenko, P. O. "A numerical algorithm for solving the generalized eigenvalue problem for completely continuous self-adjoint operators with nonlinear spectral parameter." Journal of Mathematical Sciences 88, no. 3 (February 1998): 452–55. http://dx.doi.org/10.1007/bf02365269.

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48

Bernal, Álvaro, Rafael Miró, Damián Ginestar, and Gumersindo Verdú. "Resolution of the Generalized Eigenvalue Problem in the Neutron Diffusion Equation Discretized by the Finite Volume Method." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/913043.

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Numerical methods are usually required to solve the neutron diffusion equation applied to nuclear reactors due to its heterogeneous nature. The most popular numerical techniques are the Finite Difference Method (FDM), the Coarse Mesh Finite Difference Method (CFMD), the Nodal Expansion Method (NEM), and the Nodal Collocation Method (NCM), used virtually in all neutronic diffusion codes, which give accurate results in structured meshes. However, the application of these methods in unstructured meshes to deal with complex geometries is not straightforward and it may cause problems of stability and convergence of the solution. By contrast, the Finite Element Method (FEM) and the Finite Volume Method (FVM) are easily applied to unstructured meshes. On the one hand, the FEM can be accurate for smoothly varying functions. On the other hand, the FVM is typically used in the transport equations due to the conservation of the transported quantity within the volume. In this paper, the FVM algorithm implemented in the ARB Partial Differential Equations solver has been used to discretize the neutron diffusion equation to obtain the matrices of the generalized eigenvalue problem, which has been solved by means of the SLEPc library.
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49

Xu, Pengfei, Yinjie Jia, and Mingxin Jiang. "Blind audio source separation based on a new system model and the Savitzky-Golay filter." Journal of Electrical Engineering 72, no. 3 (June 1, 2021): 208–12. http://dx.doi.org/10.2478/jee-2021-0029.

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Abstract Blind source separation (BSS) is a research hotspot in the field of signal processing. This scheme is widely applied to separate a group of source signals from a given set of observations or mixed signals. In the present study, the Savitzky-Golay filter is applied to smooth the mixed signals, adopt a simplified cost function based on the signal to noise ratio (SNR) and obtain the demixing matrix accordingly. To this end, the generalized eigenvalue problem is solved without conventional iterative methods. It is founded that the proposed algorithm has a simple structure and can be easily implemented in diverse problems. The obtained results demonstrate the good performance of the proposed model for separating audio signals in cases with high signal to noise ratios.
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50

Belyaeva, Irina, Igor Kirichenko, Oleh Ptashnyi, Natalia Chekanova, and Tetiana Yarkho. "Integrating linear ordinary fourth-order differential equations in the MAPLE programming environment." Eastern-European Journal of Enterprise Technologies 3, no. 4 (111) (June 29, 2021): 51–57. http://dx.doi.org/10.15587/1729-4061.2021.233944.

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This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.
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