A backward error for the symmetric generalized inverse eigenvalue problem

Ma, Wei

doi:10.1016/j.laa.2013.11.025

Cited by 4 publications

(3 citation statements)

References 7 publications

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“…• The derivation of Algorithm 5 can be extended to inverse generalized eigenvalue problems in [18][19][20]30], while the Cayley transform method cannot be applied to such problems.…”

Section: Algorithmmentioning

confidence: 99%

“…From the theoretical viewpoint, in 1999, Chan, Xu, and Zhou pointed out that the proof of quadratic convergence of one of the above methods [24,Theorem 3.3] was incorrect, and then gave a correct proof in [11]. In addition, backward error analysis for the inverse symmetric eigenvalue problems is given in [30,43]. As the algorithmic progress, to the best of the author's knowledge, there have been two significant developments.…”

Section: Introductionmentioning

confidence: 99%

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Formulations and theorems of quadratically convergent methods for inverse symmetric eigenvalue problems

Aishima

2020

NOLTA

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Inverse eigenvalue problems arise in a variety of applications, and thus various Newton's methods, which quadratically converge, have been developed both in theory and practice. Among many studies over thirty years, two extremely significant developments are found. Firstly, smooth matrix decompositions have been successfully applied since the 1990s. Secondly, a matrix multiplication based method has been recently proposed. In this paper, such efficient modern solvers are classified in the context of classical Newton's methods according to their mathematical formulations, and then the corresponding convergence theorems and their relationship are surveyed.

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“…• The derivation of Algorithm 5 can be extended to inverse generalized eigenvalue problems in [18][19][20]30], while the Cayley transform method cannot be applied to such problems.…”

Section: Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Formulations and theorems of quadratically convergent methods for inverse symmetric eigenvalue problems

Aishima

2020

NOLTA

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“…In , the sufficient and necessary conditions for the inverse eigenvalue problem with Hermitian, generalized skew‐Hamiltonian, and Hermitian‐generalized Hamiltonian matrices were proposed. The backward error for the symmetric generalized inverse eigenvalue problem was studied in . Recent years have seen significant interest in the development of iterative methods to solve inverse eigenvalue problems .…”

Section: Introductionmentioning

confidence: 99%

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

Hajarian

2018

Asian Journal of Control

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The inverse eigenvalue problem appears repeatedly in a variety of applications. The aim of this paper is to study a quadratic inverse eigenvalue problem of the form AXΛ2 + BXΛ + CX = 0 where A, B and C should be partially bisymmetric under a prescribed submatrix constraint. We derive an efficient matrix method based on the Hestenes‐Stiefel (HS) version of biconjugate residual (BCR) algorithm for solving this constrained quadratic inverse eigenvalue problem. The theoretical results demonstrate that the matrix method solves the constrained quadratic inverse eigenvalue problem within a finite number of iterations in the absence of round‐off errors. Finally we validate the accuracy and efficiency of the matrix method through the numerical results.

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Conjugate gradient-like algorithms for constrained operator equation related to quadratic inverse eigenvalue problems

Hajarian

2021

Comp. Appl. Math.

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A backward error for the symmetric generalized inverse eigenvalue problem

Cited by 4 publications

References 7 publications

Formulations and theorems of quadratically convergent methods for inverse symmetric eigenvalue problems

Formulations and theorems of quadratically convergent methods for inverse symmetric eigenvalue problems

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

Conjugate gradient-like algorithms for constrained operator equation related to quadratic inverse eigenvalue problems

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