The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices

Zhou, Fen; Hu, Xi-Yan; Zhang, Lei

doi:10.1016/s0024-3795(02)00550-5

Cited by 58 publications

(34 citation statements)

References 10 publications

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“…To illustrate the gain in efficiency obtained by this method, we apply the DESCM method to the time independent Schrödinger equation with an anharmonic potential. Furthermore, it is worth mentioning that research concerning inverse eigenvalue problems where the matrices are assumed centrosymmetric has been the subject of much research recently [26,31]. Consequently, the combination of these results and our findings could lead to a general approach for solving inverse Sturm-Liouville problems.…”

Section: Introductionsupporting

confidence: 61%

“…The study of centrosymmetry has a long history [11][12][13][14][15][16][17][18][19][20]. However, the last two decades has stemmed much research focused on the properties and applications of centrosymmetric matrices ranging from iterative methods for solving linear equations to least-squares problems to inverse eigenvalue problems [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

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Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Gaudreau

Safouhi

2016

EPJ Web of Conferences

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Abstract. Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1 N+1 of all components need to be computed and stored in order to obtain all eigenvalues, where 2N + 1 corresponds to the dimension of the eigensystem. We applied our result to the Schrödinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.

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Section: Introductionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Gaudreau

Safouhi

2016

EPJ Web of Conferences

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“…R n×n r (P) (R n×n a (P)) denotes the subspace reflexive (anti-reflexive) matrices with respect to the n × n generalized reflection matrix P. The reflexive and anti-reflexive matrices have practical applications in many areas such as the numerical solution of certain differential equations [1], pattern recognition [6], Markov processes [42], various physical and engineering problems [7] and so on (e.g. [20,32,43]). Chen [3] proposed three applications of reflexive and anti-reflexive matrices obtained from the altitude estimation of a level network, an electric network and structural analysis of trusses.…”

Section: Introductionmentioning

confidence: 99%

Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices

Dehghan

Hajarian

2012

Comput. Appl. Math.

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Abstract. An n × n real matrix P is said to be a generalized reflection matrix if P T = P and P 2 = I (where P T is the transpose of P). A matrix A ∈ R n×n is said to be a reflexive (antireflexive) matrix with respect to the generalized reflection matrix P if A = P A P (A = −P A P). The reflexive and anti-reflexive matrices have wide applications in many fields. In this article, two iterative algorithms are proposed to solve the coupled matrix equationsover reflexive and anti-reflexive matrices, respectively. We prove that the first (second) algorithm converges to the reflexive (anti-reflexive) solution of the coupled matrix equations for any initial reflexive (anti-reflexive) matrix. Finally two numerical examples are used to illustrate the efficiency of the proposed algorithms.Mathematical subject classification: 15A06, 15A24, 65F15, 65F20.

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“…We refer the reader to [1,16,[19][20][21][22] and references therein. For example, Zhou et al [22] and Zhang et al [21] considered the problems for the case of centrosymmetric matrices and Hermitian-generalized Hamiltonian matrices, respectively. They established existence theorems for the solutions and derived expressions of the general solutions.…”

Section: Introductionmentioning

confidence: 99%

“…We not only give an expression of the solutions but also provide a structure-preserving iterative algorithm of finite element model updating, based on the theory of inverse eigenpair problem. We also study a perturbation of the modified solution, which was not done in [20][21][22]. Next we introduce the definition of generalized centrosymmetric matrices.…”

Section: Introductionmentioning

confidence: 99%

A structure-preserving iteration method of model updating based on matrix approximation theory

Xie¹

2010

Comput. Appl. Math.

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The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices

Cited by 58 publications

References 10 publications

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices

A structure-preserving iteration method of model updating based on matrix approximation theory

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