The Inverse Eigenvector Problem for Real Tridiagonal Matrices

Parlett, Beresford Ν.; Dopico, Froilán M.; Ferreira, Carla

doi:10.1137/15m1025293

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…In [25] we show that unique tridiagonal "backward error" matrices can be designated for an approximate pair of complex eigenvectors (column and row) or two approximate real eigenvectors.…”

Section: Shift Strategymentioning

confidence: 99%

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

Ferreira

Parlett

2022

Numer. Math.

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The paper discusses the following topics: attractions of the real tridiagonal case, relative eigenvalue condition number for matrices in factored form, dqds, triple dqds, error analysis, new criteria for splitting and deflation, eigenvectors of the balanced form, twisted factorizations and generalized Rayleigh quotient iteration. We present our fast real arithmetic algorithm and compare it with alternative published approaches.

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“…In [25] we show that unique tridiagonal "backward error" matrices can be designated for an approximate pair of complex eigenvectors (column and row) or two approximate real eigenvectors.…”

Section: Shift Strategymentioning

confidence: 99%

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

Ferreira

Parlett

2022

Numer. Math.

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“…This paper focuses on the symmetric tridiagonal eigenvector problem. According to Weyl's theorem, the real symmetric eigenvalue problem Ax = xλ is well posed, in an absolute sense because an eigenvalue can change by no more than the spectral norm of the change in the matrix A [14]. However, for an unsymmetric matrix Â, some of its eigenvalues may be extremely sensitive to uncertainty in the matrix entries.…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, the assessment of error becomes a major concern. Some specific conclusions were introduced in [14]. Readers can also see more unsymmetric examples in [15,16].…”

Section: Introductionmentioning

confidence: 99%

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

2022

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This paper presents a novel method for computing the symmetric tridiagonal eigenvectors, which is the modification of the widely used Inverse Iteration method. We construct the corresponding algorithm by a new one-step iteration method, a new reorthogonalization method with the general Q iteration and a significant modification when calculating severely clustered eigenvectors. The numerical results show that this method is competitive with other existing methods, especially when computing part eigenvectors or severely clustered ones.

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“…The inverse eigenvalue problem (IEP) for linear and quadratic matrix polynomial have been well studied in the literature since the 1970s (see [17] the references therein). Some previous attempts at solving the inverse eigenvalue problem are listed in [1,23,25,39,40]. A large number of papers have been published on the linear inverse eigenvalue problem [20,41,44].…”

Section: Introductionmentioning

confidence: 99%

Solution of the Linearly Structured Partial Polynomial Inverse Eigenvalue Problem

Rakshit¹,

Khare²

2019

Preprint

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In this paper, linearly structured partial polynomial inverse eigenvalue problem is considered for the n×n matrix polynomial of arbitrary degree k. Given a set of m eigenpairs (1 m kn), this problem concerns with computing the matrices A i ∈ R n×n for i = 0, 1, 2, . . . , (k − 1) of specified linear structure such that the matrix polynomial P (λ) = λ k I n + k−1 i=0 λ i A i has the given eigenpairs as its eigenvalues and eigenvectors. Many practical applications give rise to the linearly structured structured matrix polynomial. Therefore, construction of the linearly structured matrix polynomial is the most important aspect of the polynomial inverse eigenvalue problem(PIEP). In this paper, a necessary and sufficient condition for the existence of the solution of this problem is derived. Additionally, we characterize the class of all solutions to this problem by giving the explicit expressions of solutions. The results presented in this paper address some important open problems in the area of PIEP raised in De Teran, Dopico and Van Dooren [SIAM Journal on Matrix Analysis and Applications, 36(1) (2015), pp 302 − 328]. An attractive feature of our solution approach is that it does not impose any restriction on the number of eigendata for computing the solution of PIEP. The proposed method is validated with various numerical examples on a spring mass problem.

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The Inverse Eigenvector Problem for Real Tridiagonal Matrices

Cited by 4 publications

References 12 publications

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

A real triple dqds algorithm for the nonsymmetric tridiagonal eigenvalue problem

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

Solution of the Linearly Structured Partial Polynomial Inverse Eigenvalue Problem

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