An inverse eigenvalue problem for pseudo-Jacobi matrices

Xu, Wei-Ru; Bebiano, Natália; Chen, Guoliang

doi:10.1016/j.amc.2018.10.051

Cited by 8 publications

(5 citation statements)

References 23 publications

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“…Furthermore, numerical experiments illustrate the efficiency and feasibility of the proposed construction algorithm (Algorithm 3). Our results extend the previous results obtained in [18] for the unique case when µ 1 ∩ µ 2 = ∅ as well as in [6] and [29] LEMMA A.1. Let J n ∈ J (n, , β) have distinct real eigenvalues λ 1 , λ 2 , .…”

Section: Imsupporting

confidence: 91%

“…REMARK A.2. Matrices in J (n, , β) may exist with multiple eigenvalues (see Examples 4.1 and 4.2 in [29]). It has been shown in Lemma A.1 that, if J n ∈ J (n, , β) is diagonalizable and has a multiple eigenvalue λ j , then the equality (A.1) also holds.…”

Section: Immentioning

confidence: 99%

“…The problems in this area deserve attention in order to extend the classical theory of the Jacobi case. At present, some developments focusing on pseudo-Jacobi inverse eigenvalue problem, abbreviated by the acronym PJIEP, have been obtained; see [3,4,5,6,18,21,29,30]. This paper is in the continuation of this research field.…”

mentioning

confidence: 97%

“…In [18], Mirzaei investigated the particular case of the PJIEP in which the elements of λ, µ 1 , and µ 2 are real pairwise distinct numbers and µ 1 ∩ µ 2 = ∅. The special case in which H = I r ⊕ −I 1 ⊕ I n−r−1 was solved by Xu, Bebiano, and Chen [29].…”

mentioning

confidence: 99%

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On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra

Xu¹,

Bebiano²,

Chen³

2019

etna

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Non-selfadjoint tridiagonal matrices play a role in the discretization and truncation of the Schrödinger equation in some extensions of quantum mechanics, a research field particularly active in the last two decades. In this article, we consider an inverse eigenvalue problem that consists of the reconstruction of such a real nonselfadjoint matrix from its prescribed eigenvalues and those of two complementary principal submatrices. Necessary and sufficient conditions under which the problem has a solution are presented, and uniqueness is discussed. The reconstruction is performed by using a modified unsymmetric Lanczos algorithm, designed to solve the proposed inverse eigenvalue problem. Some illustrative numerical examples are given to test the efficiency and feasibility of our reconstruction algorithm.

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Section: Imsupporting

confidence: 91%

Section: Immentioning

confidence: 99%

mentioning

confidence: 97%

mentioning

confidence: 99%

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On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra

Xu¹,

Bebiano²,

Chen³

2019

etna

Self Cite

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“…We also discuss orthogonality with respect to bilinear forms. The tridiagonal pencil IEP presented here generalizes polynomials orthogonal to a bilinear form [21] and the pseudo-Jacobi inverse eigenvalue problem [32] which arises in the study of quantum mechanics. The IEP implies short recurrence relations for rational functions orthogonal with respect to this bilinear form [1], however the solution procedure must rely on nonunitary transformations and is therefore liable to numerical instability.…”

Section: Introductionmentioning

confidence: 99%

Generation of orthogonal rational functions by procedures for structured matrices

2021

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The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

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How to choose the signature operator such that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable?

Gong

Bebiano

et al. 2022

Applied Mathematics Letters

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An inverse eigenvalue problem for pseudo-Jacobi matrices

Cited by 8 publications

References 23 publications

On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra

On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra

Generation of orthogonal rational functions by procedures for structured matrices

How to choose the signature operator such that the periodic pseudo-Jacobi inverse eigenvalue problem is solvable?

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