Journal articles: '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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Palej, Rafał, Artur Krowiak, and Renata Filipowska. "The Modified Power Method for Solving the Eigenvalue Problem with the Use of Idempotent Matrix." Applied Mechanics and Materials 712 (January 2015): 43–48. http://dx.doi.org/10.4028/www.scientific.net/amm.712.43.

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The work presents a new approach to the power method serving the purpose of solving the eigenvalue problem of a matrix. Instead of calculating the eigenvector corresponding to the dominant eigenvalue from the formula , the idempotent matrix B associated with the given matrix A is calculated from the formula , where m stands for the method’s rate of convergence. The scaling coefficient ki is determined by the quotient of any norms of matrices Bi and or by the reciprocal of the Frobenius norm of matrix Bi. In the presented approach the condition for completing calculations has the form. Once the calculations are completed, the columns of matrix B are vectors parallel to the eigenvector corresponding to the dominant eigenvalue, which is calculated from the Rayleigh quotient. The new approach eliminates the necessity to use a starting vector, increases the rate of convergence and shortens the calculation time when compared to the classic method.

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Krakoff, Benjamin, Susan M. Mniszewski, and Christian F. A. Negre. "Controlled precision QUBO-based algorithm to compute eigenvectors of symmetric matrices." PLOS ONE 17, no. 5 (2022): e0267954. http://dx.doi.org/10.1371/journal.pone.0267954.

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We describe an algorithm to compute the extremal eigenvalues and corresponding eigenvectors of a symmetric matrix which is based on solving a sequence of Quadratic Binary Optimization problems. This algorithm is robust across many different classes of symmetric matrices; It can compute the eigenvector/eigenvalue pair to essentially any arbitrary precision, and with minor modifications, can also solve the generalized eigenvalue problem. Performance is analyzed on small random matrices and selected larger matrices from practical applications.

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Schwarz, Angelika, Carl Christian Kjelgaard Mikkelsen, and Lars Karlsson. "Robust parallel eigenvector computation for the non-symmetric eigenvalue problem." Parallel Computing 100 (December 2020): 102707. http://dx.doi.org/10.1016/j.parco.2020.102707.

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DURÁN, RICARDO G., LUCIA GASTALDI, and CLAUDIO PADRA. "A POSTERIORI ERROR ESTIMATORS FOR MIXED APPROXIMATIONS OF EIGENVALUE PROBLEMS." Mathematical Models and Methods in Applied Sciences 09, no. 08 (1999): 1165–78. http://dx.doi.org/10.1142/s021820259900052x.

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In this paper we introduce and analyze an a posteriori error estimator for the approximation of the eigenvalues and eigenvectors of a second-order elliptic problem obtained by the mixed finite element method of Raviart–Thomas of the lowest order. We define an error estimator of the residual type which can be computed locally from the approximate eigenvector and prove that the estimator is equivalent to the norm of the error in the approximation of the eigenvector up to higher order terms. The constants involved in this equivalence depend on the corresponding eigenvalue but are independent of the mesh size, provided the meshes satisfy the usual minimum angle condition. Moreover, the square root of the error in the approximation of the eigenvalue is also bounded by a constant times the estimator.

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Duan, Shan-Qi, Qing-Wen Wang, and Xue-Feng Duan. "On Rayleigh Quotient Iteration for the Dual Quaternion Hermitian Eigenvalue Problem." Mathematics 12, no. 24 (2024): 4006. https://doi.org/10.3390/math12244006.

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The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the use of Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with dual representation, the RQI algorithm can effectively compute the eigenvalue along with the associated eigenvector of the dual quaternion Hermitian matrices. Furthermore, by utilizing the minimal residual property of the Rayleigh quotient, a convergence analysis of the Rayleigh quotient iteration is derived. Numerical examples are provided to illustrate the high accuracy and low CPU time cost of the proposed Rayleigh quotient iteration compared with the power method for solving the dual quaternion Hermitian eigenvalue problem.

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Ashokkumar, Chimpalthradi R., George WP York, and Scott F. Gruber. "Proportional–integral–derivative controller family for pole placement." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 20 (2016): 3791–97. http://dx.doi.org/10.1177/0954406216651893.

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In this paper, linear time-invariant square systems are considered. A procedure to design infinitely many proportional–integral–derivative controllers, all of them assigning closed-loop poles (or closed-loop eigenvalues), at desired locations fixed in the open left half plane of the complex plane is presented. The formulation accommodates partial pole placement features. The state-space realization of the linear system incorporated with a proportional–integral–derivative controller boils down to the generalized eigenvalue problem. The generalized eigenvalue-eigenvector constraint is transformed into a system of underdetermined linear homogenous set of equations whose unknowns include proportional–integral–derivative parameters. Hence, the proportional–integral–derivative solution sets are infinitely many for the chosen closed-loop eigenvalues in the eigenvalue-eigenvector constraint. The solution set is also useful to reduce the tracking errors and improve the performance. Three examples are illustrated.

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Cao, Mingyuan, Yueting Yang, Chaoqian Li, and Xiaowei Jiang. "An accelerated conjugate gradient method for the Z-eigenvalues of symmetric tensors." AIMS Mathematics 8, no. 7 (2023): 15008–23. http://dx.doi.org/10.3934/math.2023766.

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<abstract>We transform the Z-eigenvalues of symmetric tensors into unconstrained optimization problems with a shifted parameter. An accelerated conjugate gradient method is proposed for solving these unconstrained optimization problems. If solving problem results in a nonzero critical point, then it is a Z-eigenvector corresponding to the Z-eigenvalue. Otherwise, we solve the shifted problem to find a Z-eigenvalue. In our method, the new conjugate gradient parameter is a modified CD conjugate gradient parameter, and an accelerated parameter is presented by using the quasi-Newton direction. The global convergence of new method is proved. Numerical experiments are listed to illustrate the efficiency of the proposed method.</abstract>

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Deng, Quanling. "Analytical solutions to some generalized and polynomial eigenvalue problems." Special Matrices 9, no. 1 (2021): 240–56. http://dx.doi.org/10.1515/spma-2020-0135.

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Abstract It is well-known that the finite difference discretization of the Laplacian eigenvalue problem −Δu = λu leads to a matrix eigenvalue problem (EVP) Ax =λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.

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Ankerhold, Joachim, and Philip Pechukas. "Eigenvector-eigenvalue rate calculations for the fluctuating barrier problem: Two examples." Physical Review E 58, no. 6 (1998): 6968–74. http://dx.doi.org/10.1103/physreve.58.6968.

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Xie, Huiqing. "A new method for eigenvector derivatives of a quadratic eigenvalue problem." BIT Numerical Mathematics 57, no. 4 (2017): 1065–82. http://dx.doi.org/10.1007/s10543-017-0680-y.

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Cai, Yunfeng, Zhigang Jia, and Zheng-Jian Bai. "Perturbation analysis of an eigenvector-dependent nonlinear eigenvalue problem with applications." BIT Numerical Mathematics 60, no. 1 (2019): 1–29. http://dx.doi.org/10.1007/s10543-019-00765-4.

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ONOFREI, DANIEL, and BOGDAN VERNESCU. "ASYMPTOTICS OF A SPECTRAL PROBLEM ASSOCIATED WITH THE NEUMANN SIEVE." Analysis and Applications 03, no. 01 (2005): 69–87. http://dx.doi.org/10.1142/s0219530505000480.

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In this paper, we analyze the asymptotic behavior of a Stekloff spectral problem associated with the Neumann Sieve model, i.e. a three-dimensional set Ω, cut by a hyperplane Σ where each of the two-dimensional holes, ∊-periodically distributed on Σ, have diameter r∊. Depending on the asymptotic behavior of the ratios [Formula: see text] we find the limit problem of the ∊ spectral problem and prove that the sequences [Formula: see text], formed by the nth eigenvalue of the ∊ problem, converge to λn, the nth eigenvalue of the limit problem, for any n ∈ N. We also prove the weak convergence, on a subsequence, of the associated sequence of eigenvectors [Formula: see text], to an eigenvector associated with λn. When λn is a simple eigenvalue, we show that the entire sequence of the eigenvectors converges. As a consequence, similar results hold for the spectrum of the DtN map associated to this model.

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D. V. Divakov, K.P. Lovetskiy, A. L. Sevastyanov, and A. A. Tiutiunnik. "Adiabatic guided modes of a three-layer integral optical waveguide." Technical Physics 68, no. 4 (2023): 423. http://dx.doi.org/10.21883/tp.2023.04.55931.292-22.

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The numerical solution of the problem of guided propagation of polarized light in a smooth junction of a planar waveguide is considered. Within the framework of the model of adiabatic guided modes, the system of Maxwell equations is reduced to a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field in the zeroth approximation and the same number of equations in the first approximation. The multilayer structure of waveguides makes it possible to reduce the problem to a homogeneous system of linear algebraic equations, whose nontrivial solvability condition yields the dispersion equation. Auxiliary eigenvalue problems for describing the adiabatic modes of the waveguide are solved. Keywords: smoothly irregular integrated-optical multilayer waveguides, eigenvalue and eigenvector problems, single-mode propagation of adiabatic waveguide modes.

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Qiu, Yuyang. "Eigenvector-Free Solutions to the Matrix EquationAXBH=Ewith Two Special Constraints." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/869705.

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The matrix equationAXBH=EwithSX=XRorPX=sXQconstraint is considered, whereS, Rare Hermitian idempotent,P, Qare Hermitian involutory, ands=±1. By the eigenvalue decompositions ofS, R, the equationAXBH=EwithSX=XRconstraint is equivalently transformed to an unconstrained problem whose coefficient matrices contain the corresponding eigenvectors, with which the constrained solutions are constructed. The involved eigenvectors are released by Moore-Penrose generalized inverses, and the eigenvector-free formulas of the general solutions are presented. By choosing suitable matricesS, R, we also present the eigenvector-free formulas of the general solutions to the matrix equationAXBH=EwithPX=sXQconstraint.

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Bazilevskiy, M. P. "Solving an Optimization Problem for Estimating Fully connected Linear Regression Models." Моделирование и анализ данных 14, no. 1 (2024): 121–34. http://dx.doi.org/10.17759/mda.2024140108.

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This article is devoted to the problem of estimating fully connected linear regression models using the maximum likelihood method. Previously, a special numerical method was developed for this purpose, based on solving a nonlinear system using the method of simple iterations. At the same time, the issues of choosing initial approximations and fulfilling sufficient conditions for convergence were not studied. This article proposes a new method for solving the optimization problem of estimating fully connected regressions, similar to the method of estimating orthogonal regressions. It has been proven that, with equal error variances of interconnected variables, estimates of b-parameters of fully connected regression are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse covariance matrix. And if the ratios of the error variances of the variables are equal to the ratios of the variances of the variables, then the b-parameter estimates are equal to the components of the eigenvector corresponding to the smallest eigenvalue of the inverse correlation matrix, multiplied by the specific ratios of the standard deviations of the variables. A numerical experiment was carried out to confirm the correctness of the developed mathematical apparatus. The proposed method for solving the optimization problem of estimating fully connected regressions can be effectively used when solving problems of constructing multiple fully connected linear regressions.

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de Macêdo Wahrhaftig, Alexandre. "Analysis of the First Modal Shape Using Two Case Studies." International Journal of Computational Methods 16, no. 06 (2019): 1840019. http://dx.doi.org/10.1142/s0219876218400194.

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Eigenvector analysis can be performed to determine the shapes of the undamped free vibration modes of a system. Eigenvector analysis involves solving the generalized eigenvalue problem, which considers the stiffness and mass matrix of a structure. For a geometric nonlinear study, both parts of the total stiffness matrix are required. As modal analysis depends on the stiffness, the effect of its reduction on the modal shape of vibration of the structure must be determined. Case studies were evaluated using the finite element method, considering and neglecting the geometric portion of the stiffness matrix. Mathematic functions were applied for comparison.

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YANG, B. J. "DOMINANT EIGENVECTOR AND EIGENVALUE ALGORITHM IN SPARSE NETWORK SPECTRAL CLUSTERING." Latin American Applied Research - An international journal 48, no. 4 (2018): 323–28. http://dx.doi.org/10.52292/j.laar.2018.248.

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The sparse network spectrum clustering problem is studied in this paper. It tries to analyze and improve the sparse network spectrum clustering algorithm from the main feature pair algorithm. The main feature pair algorithm in the matrix calculation is combined with the spectral clustering algorithm to explore the application of the main feature pair algorithm on the network adjacency matrix. The defects of traditional main features are analyzed when the algorithm Power is used on the network of special structural features, and the advantages of the new algorithm SII algorithm is proved. The sparse network spectral clustering algorithm in this paper is based on the Score algorithm, and the main features of the algorithm are refined, analyzed and improved.

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Maghribi, Sahmura Maula Al, Siswanto Siswanto, and Sutrima Sutrima. "Characteristic Min-Polynomial and Eigen Problem of a Matrix over Min-Plus Algebra." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 4 (2023): 1108. http://dx.doi.org/10.31764/jtam.v7i4.16498.

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Let R_ε=R∪{-∞}, with R being a set of all real numbers. The algebraic structure (R_ε,⊕,⊗) is called max-plus algebra. The task of finding the eigenvalue and eigenvector is called the eigenproblem. There are several methods developed to solve the eigenproblem of A∈R_ε^(n×n), one of them is by using the characteristic max-polynomial. There is another algebraic structure that is isomorphic with max-plus algebra, namely min-plus algebra. Min-plus algebra is a set of R_(ε^' )=R∪{+∞} that uses minimum (⊕^' ) and addition (⊗) operations. The eigenproblem in min-plus algebra is to determine λ∈R_(ε^' ) and v∈R_(ε^')^n such that A⊗v=λ⊗v. In this paper, we provide a method for determining the characteristic min-polynomial and solving the eigenproblem by using the characteristic min-polynomial. We show that the characteristic min-polynomial of A∈R_(ε^')^(n×n) is the permanent of I⊗x⊕^' A, the smallest corner of χ_A (x) is the principal eigenvalue (λ(A)), and the columns of A_λ^+ with zero diagonal elements are eigenvectors corresponding to the principal eigenvalue.

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Lovetsky, K. P., L. A. Sevastianov, and N. P. Tretiakov. "An Exact Penalty Function Method for Solving the Full Eigenvalue and Eigenvector Problem." Journal of Computational Methods in Sciences and Engineering 2, no. 1-2 (2002): 189–94. http://dx.doi.org/10.3233/jcm-2002-21-226.

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McCartin, Brian J. "Pseudoinverse formulation of Rayleigh-Schrödinger perturbation theory for the symmetric matrix eigenvalue problem." Journal of Applied Mathematics 2003, no. 9 (2003): 459–85. http://dx.doi.org/10.1155/s1110757x03303092.

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A comprehensive treatment of Rayleigh-Schrödinger perturbation theory for the symmetric matrix eigenvalue problem is furnished with emphasis on the degenerate problem. The treatment is simply based upon the Moore-Penrose pseudoinverse thus distinguishing it from alternative approaches in the literature. In addition to providing a concise matrix-theoretic formulation of this procedure, it also provides for the explicit determination of that stage of the algorithm where each higher-order eigenvector correction becomes fully determined. The theory is built up gradually with each successive stage appended with an illustrative example.

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Khademloo, S. "A Multiplicity Result for Quasilinear Problems with Nonlinear Boundary Conditions in Bounded Domains." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/419341.

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We study the following quasilinear problem with nonlinear boundary condition , in and on , where is a connected bounded domain with smooth boundary , the outward unit normal to which is denoted by . is the -Laplcian operator defined by , the functions and are sign changing continuous functions in , , where if and otherwise. The properties of the first eigenvalue and the associated eigenvector of the related eigenvalue problem have been studied in (Khademloo, In press). In this paper, it is shown that if , the original problem admits at least one positive solution, while if , for a positive constant , it admits at least two distinct positive solutions. Our approach is variational in character and our results extend those of Afrouzi and Khademloo (2007) in two aspects: the main part of our differential equation is the -Laplacian, and the boundary condition in this paper also is nonlinear.

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Sciriha, Irene. "Joining Forces for Reconstruction Inverse Problems." Symmetry 13, no. 9 (2021): 1687. http://dx.doi.org/10.3390/sym13091687.

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A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of parameters to determine a graph uniquely. Other related inverse problems, including the Polynomial Reconstruction Problem (PRP), involve the recovery of graph invariants. The PRP seeks to extract the spectrum of a graph from the deck of cards each showing the spectrum of a vertex-deleted subgraph. We show how various algebraic methods join forces to reconstruct a graph or its invariants from a minimal set of restricted eigenvalue-eigenvector information of the parent graph or its subgraphs. We show how functions of the entries of eigenvectors of the adjacency matrix A of a graph can be retrieved from the spectrum of eigenvalues of A. We establish that there are two subclasses of disconnected graphs with each card of the deck showing a common eigenvalue. These could occur as possible counter examples to the positive solution of the PRP.

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Ábele-Nagy, Kristóf. "Minimization of the Perron eigenvalue of incomplete pairwise comparison matrices by Newton iteration." Acta Universitatis Sapientiae, Informatica 7, no. 1 (2015): 58–71. http://dx.doi.org/10.1515/ausi-2015-0012.

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Abstract Pairwise comparison matrices are of key importance in multi-attribute decision analysis. A matrix is incomplete if some of the elements are missing. The eigenvector method, to derive the weights of criteria, can be generalized for the incomplete case by using the least inconsistent completion of the matrix. If inconsistency is indexed by CR, defined by Saaty, it leads to the minimization of the Perron eigenvalue. This problem can be transformed to a convex optimization problem. The paper presents a method based on the Newton iteration, univariate and multivariate. Numerical examples are also given.

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Fenu, Luigi, Eleonora Congiu, Mariangela Deligia, Gian Felice Giaccu, Alireza Hosseini, and Mauro Serra. "Buckling Analysis of Piles in Multi-Layered Soils." Applied Sciences 11, no. 22 (2021): 10624. http://dx.doi.org/10.3390/app112210624.

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Pile buckling is infrequent, but sometimes it can occur in slender piles (i.e., micropiles) driven into soils with soft layers and/or voids. Buckling analysis of piles becomes more complex if the pile is surrounded by multi-layered soil. In this case, the well-known Timoshenko’s solution for pile buckling is of no use because it refers to single-layered soils. A variational approach for buckling analysis of piles in multi-layered soils is herein proposed. The proposed method allows for the estimation of the critical buckling load of piles in any multi-layered soil and for any boundary condition, provided that the distribution of the soil coefficient of the subgrade reaction is available. An eigenvalue-eigenvector problem is defined, where each eigenvector is the set of coefficients of a Fourier series describing the second-order displaced shape of the pile, and the related buckling load is the eigenvalue, thus obtaining the effective buckling load as the minimum eigenvalue. Besides the pile deformed shape, the stiffness distribution in the multi-layered soil is also described through a Fourier series. The Rayleigh–Ritz direct method is used to identify the Fourier development coefficients describing the pile deformation. For validation, buckling analysis results were compared with those obtained from an experimental test and a finite element analysis available in the literature, which confirmed this method’s reliability.

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Liu, Chein-Shan, Chung-Lun Kuo, and Chih-Wen Chang. "Free Vibrations of Multi-Degree Structures: Solving Quadratic Eigenvalue Problems with an Excitation and Fast Iterative Detection Method." Vibration 5, no. 4 (2022): 914–35. http://dx.doi.org/10.3390/vibration5040053.

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For the free vibrations of multi-degree mechanical structures appeared in structural dynamics, we solve the quadratic eigenvalue problem either by linearizing it to a generalized eigenvalue problem or directly treating it by developing the iterative detection methods for the real and complex eigenvalues. To solve the generalized eigenvalue problem, we impose a nonzero exciting vector into the eigen-equation, and solve a nonhomogeneous linear system to obtain a response curve, which consists of the magnitudes of the n-vectors with respect to the eigen-parameters in a range. The n-dimensional eigenvector is supposed to be a superposition of a constant exciting vector and an m-vector, which can be obtained in terms of eigen-parameter by solving the projected eigen-equation. In doing so, we can save computational cost because the response curve is generated from the data acquired in a lower dimensional subspace. We develop a fast iterative detection method by maximizing the magnitude to locate the eigenvalue, which appears as a peak in the response curve. Through zoom-in sequentially, very accurate eigenvalue can be obtained. We reduce the number of eigen-equation to n−1 to find the eigen-mode with its certain component being normalized to the unit. The real and complex eigenvalues and eigen-modes can be determined simultaneously, quickly and accurately by the proposed methods.

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Altmann, Robert, Patrick Henning, and Daniel Peterseim. "The J-method for the Gross–Pitaevskii eigenvalue problem." Numerische Mathematik 148, no. 3 (2021): 575–610. http://dx.doi.org/10.1007/s00211-021-01216-5.

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AbstractThis paper studies the J-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross–Pitaevskii eigenvalue problem, we prove global convergence towards an eigenfunction for a damped version of the J-method. More importantly, when the iterations are sufficiently close to an eigenfunction, the damping can be switched off and we recover a local linear convergence rate previously known from the discrete setting. This quantitative convergence analysis is closely connected to the J-method’s unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows, this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps.

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Aleksey, Anatolievich Zakharenko. "The problem of finding of eigenvectors for 4P-SH-SAW propagation in 6 mm media." Canadian Journal of Pure and Applied Sciences 11, no. 1 (2017): 4103–19. https://doi.org/10.5281/zenodo.1301202.

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This theoretical report is pertinent to the mathematical problem of finding of all the possible eigenvectors for the four-potential shear-horizontal surface acoustic wave (4P-SH-SAW) propagation in suitable solids. In this case, the wave propagation is coupled with the four potentials, i.e. the electrical, magnetic, gravitational, and cogravitational ones. The taking into account these four potentials results in significant difficulties to find any eigenvector because the mathematical method is significantly complicated. To find all suitable eigenvectors is very important here because it will allow one in the future to theoretically disclose all suitable solutions of acoustic waves. This is applicable to the problem of finding of propagation velocities of the SH-SAWs, interfacial SH-waves, plate SH-waves, and more complicated cases. It is thought that all the effects (for instance, the gravitocogravitic, gravitoelectric, cogravitoelectric, gravitomagnetic, cogravitomagnetic effects) individually or collaboratively participating in the acoustic wave propagation can be vital for acoustic wave propagation that can be readily used for constitution of suitable technical devices. This fact must be first demonstrated theoretically for experimentalists and engineers working with the transmitting, detecting, and converting of the electromagnetic waves' signals. It is expected that the future communication technologies will also exploit gravitational waves for the new communication era based on some gravitational phenomena.

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Pavluš, Miron, Rostislav Tomeš, and Lukáš Malec. "Two Proofs and One Algorithm Related to the Analytic Hierarchy Process." Journal of Applied Mathematics 2018 (December 2, 2018): 1–9. http://dx.doi.org/10.1155/2018/5241537.

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36 years ago, Thomas Saaty introduced a new mathematical methodology, called Analytic Hierarchy Process (AHP), regarding the decision-making processes. The methodology was widely applied by Saaty and by other authors in the different human activity areas, like planning, business, education, healthcare, etc. but, in general, in the area of management. In this paper, we provide two new proofs for well-known statement that the maximal eigenvalue λmax is equal to n for the eigenvector problem Aw=λw, where A is, so-called, the consistent matrix of pairwise comparisons of type n×n (n≥ 2) with the solution vector w that represents the probability components of disjoint events. Moreover, we suggest an algorithm for the determination of the eigenvalue problem solution Aw=nw as well as the corresponding flowchart. The algorithm for arbitrary consistent matrix A can be simply programmed and used.

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Truhar, Ninoslav, and Ren-Cang Li. "On an eigenvector-dependent nonlinear eigenvalue problem from the perspective of relative perturbation theory." Journal of Computational and Applied Mathematics 395 (October 2021): 113596. http://dx.doi.org/10.1016/j.cam.2021.113596.

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Aliev, Rafik A., Witold Pedrycz, Oleg H. Huseynov, and Rafig R. Aliyev. "Eigensolutions of Partially Reliable Decision Preferences Described by Matrices of Z-Numbers." International Journal of Information Technology & Decision Making 19, no. 06 (2020): 1429–50. http://dx.doi.org/10.1142/s0219622020500340.

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Eigenvalues and eigenvectors are widely used in various applications. Particularly, these concepts underlie analysis of consistency of a decision maker’s (DMs) preference knowledge. In real-world problems, DMs knowledge is inherently associated with imprecision and partial reliability. This involves combination of fuzzy and probabilistic information. The concept of a Z-number is a formal construct to describe such kind of information. In this study, we formulate the concepts of Z-number valued eigenvalue and eigenvector for matrices components of which are Z-numbers. A formal statement of the problem and a solution method for computation of Z-number valued eigensolutions are proposed. Numerical examples and an application devoted to foreign market selection problem are provided to show the usefulness of the proposed approach.

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Yang, Ming, Xin Xiang Zhou, De Chen Zhang, Xiu E. Wu, and Xi Chen. "Study of Vehicle Vibration Model Based on Lagrange Method." Applied Mechanics and Materials 328 (June 2013): 585–88. http://dx.doi.org/10.4028/www.scientific.net/amm.328.585.

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The vibrating behavior of multiple DOF vehicle systems is very much dependent to their natural frequencies and mode shapes. These characteristics can be determined by solving an eigenvalue and an eigenvector problem. The kinetic energy, potential energy, and dissipation function of the system was defined by quadratures and the equations of motion were derived by applying the Lagrange method. Numerical example was calculated. The results show that the method is correct and it lay a good foundation for further research on vehicle vibration.

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CHIAPPINELLI, RAFFAELE, MASSIMO FURI, and MARIA PATRIZIA PERA. "PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE." Glasgow Mathematical Journal 55, no. 3 (2013): 629–38. http://dx.doi.org/10.1017/s0017089512000791.

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AbstractLet H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax + ε B(x) =δ x, where A: H → H is a bounded self-adjoint (linear) operator with nontrivial kernel Ker A, and B: H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x0 ∈ S∩ Ker A of A (thus corresponding to the eigenvalue δ=0, which we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S∩ Ker A), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ ℝ and ε ≠ 0. In this paper, we prove that if B is a C1 gradient mapping and the eigenvalue δ=0 has finite multiplicity, then the sphere S∩ Ker A contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.

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Haney, Matthew M., and Victor C. Tsai. "Perturbational and nonperturbational inversion of Love-wave velocities." GEOPHYSICS 85, no. 1 (2019): F19—F26. http://dx.doi.org/10.1190/geo2018-0882.1.

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We describe a set of MATLAB codes to forward model and invert Love-wave phase or group velocities. The forward modeling is based on a finite-element method in the frequency-wavenumber domain, and we obtain the different modes with an eigenvector-eigenvalue solver. We examine the issue of parasitic modes that arises for modeling Love waves, in contrast to the Rayleigh wave case, and how to discern parasitic from physical modes. Once the matrix eigenvector-eigenvalue problem has been solved for Love waves, we show a straightforward technique to obtain sensitivity kernels for S-wave velocity and density. In practice, the sensitivity of Love waves to density is relatively small and inversions only aim to estimate the S-wave velocity. Two types of inversion accompany the forward-modeling codes: One is a perturbational scheme for updating an initial model, and the other is a nonperturbational method that is well-suited for defining a good initial model. The codes are able to implement an optimal nonuniform layering designed for Love waves, invert combinations of phase and group velocity measurements of any mode, and seamlessly handle the transition from guided to leaky modes below the cutoff frequency. Two software examples demonstrate use of the codes at near-surface and crustal scales.

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Suarez, Luis E., and Mahendra P. Singh. "Successive Synthesis of Substructure Modes." Journal of Applied Mechanics 58, no. 3 (1991): 759–65. http://dx.doi.org/10.1115/1.2897261.

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A mode synthesis approach is presented to calculate the eigenproperties of a structure from the eigenproperties of its substructures. The approach consists of synthesizing the substructures sequentially, one degree-of-freedom at a time. At each coupling stage, the eigenvalue is obtained as the solution of a characteristic equation, defined in closed form in terms of the eigenproperties obtained in the preceding coupling stage. The roots of the characteristic equation can be obtained by a simple Newton-Raphson root finding scheme. For each calculated eigenvalue, the eigenvector is defined by a simple closed-form expression. The eigenproperties obtained in the final coupling stage provide the desired eigenproperties of the coupled system. Thus, the approach avoids a conventional solution of the second eigenvalue problem. The approach can be implemented with the complete set or a truncated number of substructure modes; if the complete set of modes is used, the calculated eigenproperties would be exact. The approach can be used with any finite element discretization of structures. It requires only the free interface eigenproperties of the substructures. Successful application of the approach to a moderate size problem (255 degrees-of-freedom) on a microcomputer is also demonstrated.

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Liu, Xinyu, Jin Zhang, Haibo Ji, and Xinghu Wang. "Distributed Nash equilibrium seeking design for energy consumption games of HVAC systems over digraphs." Journal of University of Science and Technology of China 52, no. 1 (2022): 5. http://dx.doi.org/10.52396/justc-2021-0153.

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The energy consumption problem of heating, ventilation, and air conditioning systems over general directed graphs is investigated. The considered problem is firstly reformulated as a Nash equilibrium seeking problem, and a distributed consensus-based algorithm is then proposed to solve it. To address the challenge arising from general directed graphs, a distributed estimation algorithm is embedded such that the explicit dependence on the left eigenvector associated with the eigenvalue zero of the Laplacian matrix can be avoided. Then, the exponential convergence of the proposed distributed Nash equilibrium seeking algorithm is established under a standing assumption. A numerical example is finally provided to verify the effectiveness of the proposed algorithm.

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Javed, Saira. "Natural Frequencies Calculation of Composite Annular Circular Plates with Variable Thickness Using the Spline Method." Journal of Composites Science 6, no. 3 (2022): 70. http://dx.doi.org/10.3390/jcs6030070.

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The present study adds to the knowledge of the free vibration of antisymmetric angle-ply annular circular plates with variable thickness for simply supported boundary conditions. The differential equations in terms of displacement and rotational functions are approximated using cubic spline approximation. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The vibration of the annular circular plates is examined for circumferential node number, radii ratio, different thickness variations, number of layers, stacking sequences and lamination materials.

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Rasekhmanesh, Mohamad Hosein, Gines Garcia-Contreras, Juan Córcoles, and Jorge A. Ruiz-Cruz. "On the Use of Quadrilateral Meshes for Enhanced Analysis of Waveguide Devices with Manhattan-Type Geometry Cross-Sections." Mathematics 10, no. 4 (2022): 656. http://dx.doi.org/10.3390/math10040656.

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This work addresses the suitability of using structured meshes composed of quadrilateral finite elements, instead of the classical unstructured meshes made of triangular elements. These meshes are used in the modal analysis of waveguides with Manhattan-like cross-sections. For this problem, solved with the two-dimensional Finite Element Method, there are two main quality metrics: eigenvalue and eigenvector accuracy. The eigenvalue accuracy is first considered, showing how the proposed structured meshes are, given comparable densities, better, especially when dealing with waveguides presenting pairs of modes with the same cutoff frequency. The second metric is measured through a practical problem, which commonly appears in microwave engineering: discontinuity analysis. In this problem, for which the Mode-Matching technique is used, eigenvectors are needed to compute the coupling between the modes in the discontinuities, directly influencing the quality of the transmission and reflection parameters. In this case, it is found that the proposed analysis performs better given low-density meshes and mode counts, thus proving that quadrilateral-element structured meshes are more resilient than their triangular counterparts to higher-order eigenvectors.

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Höllwieser, Roman, Francesco Knechtli, and Mike Peardon. "The static potential using trial states from Laplacian eigenmodes." EPJ Web of Conferences 274 (2022): 02008. http://dx.doi.org/10.1051/epjconf/202227402008.

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We compute the static potential of a quark-antiquark pair in lattice QCD using a method which is not based on Wilson loops, but where the trial states are formed by eigenvector components of the covariant lattice Laplace operator. The computational effort of this method is significantly lower than the standard Wilson loop calculation, when computing the static potential not only for on-axis, but also for many off-axis quark-antiquark separations, i.e., when a fine spatial resolution is required, e.g., for string breaking calculations. We further improve the signal by using multiple eigenvector pairs, weighted with Gaussian profile functions of the eigenvalues, providing a basis for a generalized eigenvalue problem (GEVP), as it was recently introduced to improve distillation in meson spectroscopy. We show results from the new method for the static potential with dynamical fermions and demonstrate its efficiency compared to traditional Wilson loop calculations.

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Nwachukwu, Ikenna Marcel, Ogbonna Onwuka David, Sylvia Onwuka Ulari, C. Njoku F., M. Ibearugbulem O., and C. Onyechere I. "TORSIONAL-FLEXURAL BUCKLING OF THIN-WALL COLUMN OF SINGLE SYMMETRICAL OPEN SECTION USING RITZ METHOD." International Journal of Civil and Structural Engineering Research 12, no. 2 (2024): 59–74. https://doi.org/10.5281/zenodo.14410649.

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Abstract: This paper investigates the torsional-flexural buckling behavior of thin-walled columns with single symmetric open sections. It focuses on deriving and solving the buckling equations for these specific column configurations. The analysis reduces the problem to a system of algebraic eigenvalue-eigenvector problems, identifying the critical buckling loads and modes. The buckling behavior is described by a system of three homogeneous differential equations, with two uncoupled equations, simplifying the analysis. Numerical examples illustrate that critical buckling loads decrease as column length increases, highlighting the relationship between length and stability. The results were validated through comparisons with established methods, including the differential equations method by Jerath (2020) and the equilibrium of deformed shape approach by Iyengar (1988), both of which show consistent results. This research contributes to a deeper understanding of the stability of thin-walled columns, providing essential insights for structural design and safety. Keywords: Single symmetric section, thin-walled column, flexural-torsional buckling, Ritz method, eigenvalue-eigenvector problem. Title: TORSIONAL-FLEXURAL BUCKLING OF THIN-WALL COLUMN OF SINGLE SYMMETRICAL OPEN SECTION USING RITZ METHOD Author: Nwachukwu Ikenna Marcel, David Ogbonna Onwuka, Ulari Sylvia Onwuka, F. C. Njoku, O. M. Ibearugbulem, I. C. Onyechere International Journal of Civil and Structural Engineering Research   ISSN 2348-7607 (Online) Vol. 12, Issue 2, October 2024 - March 2025 Page No: 59-74 Research Publish Journals Website: www.researchpublish.com Published Date: 12-December-2024 DOI: https://doi.org/10.5281/zenodo.14410649 Paper Download Link (Source) https://www.researchpublish.com/papers/torsional-flexural-buckling-of-thin-wall-column-of-single-symmetrical-open-section-using-ritz-method

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Shen, Qin-Qin, and Quan Shi. "Inexact modified positive-definite and skew-Hermitian splitting preconditioners for generalized saddle point problems." Advances in Mechanical Engineering 10, no. 10 (2018): 168781401880409. http://dx.doi.org/10.1177/1687814018804092.

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In this article, to better implement the modified positive-definite and skew-Hermitian splitting preconditioners studied recently (Numer. Algor., 72 (2016) 243–258) for generalized saddle point problems, a class of inexact modified positive-definite and skew-Hermitian splitting preconditioners is proposed with improved computing efficiency. Some spectral properties, including the eigenvalue distribution, the eigenvector distribution, and an upper bound of the degree of the minimal polynomial of the inexact modified positive-definite and skew-Hermitian splitting preconditioned matrices are studied. In addition, a theoretical optimal inexact modified positive-definite and skew-Hermitian splitting preconditioner is obtained. Numerical experiments arising from a model steady incompressible Navier–Stokes problem are used to validate the theoretical results and illustrate the effectiveness of this new class of proposed preconditioners.

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Journal articles on the topic 'Eigenvalue-eigenvector problem'

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