A best upper bound for the 2-norm condition number of a matrix

Merikoski, Jorma K.; Urpala, Uoti; Virtanen, Ari; Tam, Tin-Yau; Uhlig, F.

doi:10.1016/s0024-3795(96)00474-0

Cited by 32 publications

(9 citation statements)

References 2 publications

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“…The bound on the error (16) depends on the condition number of V , which is defined as cond(V ) := V −1 |||V |||. An upper bound on the condition number of matrices is widely studied, see [20], however, to our knowledge, there is no upper bound on the condition number of eigenvector matrix V . Therefore, in general, to minimize |||N F J||| by a suitable choice of N doesn't ensure the minimization of the bound (16) in the case of design 1 (Proposition IV.2).…”

Section: A Boundedness Of the Estimation Errormentioning

confidence: 99%

Scale-Free Estimation of the Average State in Large-Scale Systems

Niazi

Deplano

Canudas‐de‐Wit

et al. 2020

IEEE Control Syst. Lett.

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This paper provides a computationally tractable necessary and sufficient condition for the existence of an average state observer for large-scale linear time-invariant (LTI) systems. Two design procedures, each with its own significance, are proposed. When the necessary and sufficient condition is not satisfied, a methodology is devised to obtain an optimal asymptotic estimate of the average state. In particular, the estimation problem is addressed by aggregating the unmeasured states of the original system and obtaining a projected system of reduced dimension. This approach reduces the complexity of the estimation task and yields an observer of dimension one. Moreover, it turns out that the dimension of the system also does not affect the upper bound on the estimation error.

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Section: A Boundedness Of the Estimation Errormentioning

confidence: 99%

Scale-Free Estimation of the Average State in Large-Scale Systems

Niazi

Deplano

Canudas‐de‐Wit

et al. 2020

IEEE Control Syst. Lett.

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“…By the matrix-determinant lemma , we can obtain as (8) where is an submatrix of . Substituting (8) in (6), we can obtain as (9) To obtain determinants in (9), we decompose matrix as (10) where is an matrix of ones. Since is of unity rank, it has only a single nonzero eigenvalue whereas other eigenvalues are zero.…”

Section: B Zf Crosstalk Canceler/precoder For Vectored Vdslmentioning

confidence: 99%

“…From (12), we can see that the relative error in the determinant can be very small due to the well conditioned nature of the DSL channel. To show this, we obtain an upper bound on the 2-norm condition number as [8] (13)…”

Section: Appendix Validation Of Approximationmentioning

confidence: 99%

Performance Analysis of Zero Forcing Crosstalk Canceler in Vectored VDSL2

Zafaruddin

Prakriya

Prasad

2012

IEEE Signal Process. Lett.

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In this letter, we analyze the performance of a zero forcing (ZF) crosstalk canceler/precoder in vectored VDSL2. We show that there is gap between the theoretical capacity and the performance of the ZF canceler/precoder at higher frequencies and/or large number of vectored users. Unlike bounds developed to date, the tight rate approximations presented in this paper are valid over all frequency bands, even when the number of vectored users is large. Numerical results confirm the tightness of the derived approximations.

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“…2 is not occasional. See the following inequality [25], [26]: (6) The equality is due to the unit-norm constraints. From 6…”

Section: Determinant and The Condition Numbermentioning

confidence: 99%

Nonorthogonal Approximate Joint Diagonalization With Well-Conditioned Diagonalizers

Zhou

Xie

Yang

et al. 2009

IEEE Trans. Neural Netw.

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To make the results reasonable, existing joint diagonalization algorithms have imposed a variety of constraints on diagonalizers. Actually, those constraints can be imposed uniformly by minimizing the condition number of diagonalizers. Motivated by this, the approximate joint diagonalization problem is reviewed as a multiobjective optimization problem for the first time. Based on this, a new algorithm for nonorthogonal joint diagonalization is developed. The new algorithm yields diagonalizers which not only minimize the diagonalization error but also have as small condition numbers as possible. Meanwhile, degenerate solutions are avoided strictly. Besides, the new algorithm imposes few restrictions on the target set of matrices to be diagonalized, which makes it widely applicable. Primary results on convergence are presented and we also show that, for exactly jointly diagonalizable sets, no local minima exist and the solutions are unique under mild conditions. Extensive numerical simulations illustrate the performance of the algorithm and provide comparison with other leading diagonalization methods. The practical use of our algorithm is shown for blind source separation (BSS) problems, especially when ill-conditioned mixing matrices are involved.

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A best upper bound for the 2-norm condition number of a matrix

Cited by 32 publications

References 2 publications

Scale-Free Estimation of the Average State in Large-Scale Systems

Scale-Free Estimation of the Average State in Large-Scale Systems

Performance Analysis of Zero Forcing Crosstalk Canceler in Vectored VDSL2

Nonorthogonal Approximate Joint Diagonalization With Well-Conditioned Diagonalizers

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