On perfect conditioning of Vandermonde matrices on the unit circle

Berman, Lihu; Feuer, Arie

doi:10.13001/1081-3810.1190

Cited by 21 publications

(19 citation statements)

References 12 publications

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“…In fact, if the λ i 's are on the unit circle, then κ 2 (V m ) is, under some additional assumptions (e.g. that the nodes are not tightly clustered), usually moderate [3], [4]. If the underlying operator is nearly unitary then the λ i 's will be close to the unit circle and V m is usually well conditioned.…”

Section: Computation With Vandermonde Matricesmentioning

confidence: 99%

Data Driven Koopman Spectral Analysis in Vandermonde--Cauchy Form via the DFT: Numerical Method and Theoretical Insights

Drmač¹,

Mezić²,

Mohr³

2019

SIAM J. Sci. Comput.

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The goals and contributions of this paper are twofold. It provides a new computational tool for data driven Koopman spectral analysis by taking up the formidable challenge to develop a numerically robust algorithm by following the natural formulation via the Krylov decomposition with the Frobenius companion matrix, and by using its eigenvectors explicitly -these are defined as the inverse of the notoriously ill-conditioned Vandermonde matrix. The key step to curb ill-conditioning is the discrete Fourier transform of the snapshots; in the new representation, the Vandermonde matrix is transformed into a generalized Cauchy matrix, which then allows accurate computation by specially tailored algorithms of numerical linear algebra. The second goal is to shed light on the connection between the formulas for optimal reconstruction weights when reconstructing snapshots using subsets of the computed Koopman modes. It is shown how using a certain weaker form of generalized inverses leads to explicit reconstruction formulas that match the abstract results from Koopman spectral theory, in particular the Generalized Laplace Analysis.AMS subject classifications: 15A12, 15A23, 65F35, 65L05, 65M20, 65M22, 93A15, 93A30, 93B18, 93B40, 93B60, 93C05, 93C10, 93C15, 93C20, 93C57In this paper, we revisit the natural formulation of finite dimensional Koopman spectral analysis in terms of Krylov bases and the Frobenius companion matrix. In this formulation, reviewed in §2.1 below, a spectral approximation of A is obtained by the Rayleigh-Ritz extraction using the primitive Krylov basis X m = (f 1 , . . . , f m ). This means that the Rayleigh quotient of A is the Frobenius companion matrix, whose eigenvector matrix is the inverse of the Vandermonde matrix V m , parametrized by its eigenvalues λ i . The DMD (Koopman) modes, that is, the Ritz vectors z i , are then computed as. . , m. Unfortunately, Vandermonde matrices can have extremely high conditions numbers, so a straightforward implementation of this algebraically elegant scheme can lead to inaccurate results in finite precision computation. Most other DMD-variants bypass this issue by computing an orthonormal basis from the snaphshots using a truncated singular value decomposition [34,33,38]. We note that the original formulation of DMD was based on the SVD ([34], reviewed in Algorithm 2.1 in this paper), whereas the first connection between DMD and Koopman operator theory was formulated in terms of the companion matrix [32]. In Rowley et al. [32], though, there was no consideration of the deep numerical issues related to working with Vandermonde matrices which has likely contributed to the prevalence of the SVD-based variants of DMD. There is, however, a certain intrinsic elegance in the decomposition of the snapshots in terms of the spectral structure of the companion matrix in addition to a stronger connection to Generalized Laplace Analysis (GLA) theory [27,26] of Koopman operator theory than the SVD-based DMD variants have.Following this natural formulation, we present a DMD alg...

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Section: Computation With Vandermonde Matricesmentioning

confidence: 99%

Data Driven Koopman Spectral Analysis in Vandermonde--Cauchy Form via the DFT: Numerical Method and Theoretical Insights

Drmač¹,

Mezić²,

Mohr³

2019

SIAM J. Sci. Comput.

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“…Computing κ(F RF ) is a challenge, especially because the analysis must be valid for the entire range of antennas. Vandermonde matrices with positive real nodes z k ∈ R + are well-known to be ill-conditioned [23] -the condition number grows at least exponentially with the number of nodes K. However, if the nodes are complex-valued, it is possible to lower this growth to polynomial [24] and even achieve perfect conditioning choosing the nodes to be roots of unity [25]. In [26], the authors generalize this result to nodes that are close enough to the unit circle (not necessarily on the unit circle) and not so close to each other, while having N large enough.…”

Section: A Transmitted Powermentioning

confidence: 99%

Optimal design of energy-efficient millimeter wave hybrid transceivers for wireless backhaul

Pizzo

Sanguinetti

2017

2017 15th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)

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This work analyzes a mmWave single-cell network, which comprises a macro base station (BS) and an overlaid tier of small-cell BSs using a wireless backhaul for data traffic. We look for the optimal number of antennas at both BS and smallcell BSs that maximize the energy efficiency (EE) of the system when a hybrid transceiver architecture is employed. Closed-form expressions for the EE-optimal values of the number of antennas are derived that provide valuable insights into the interplay between the optimization variables and hardware characteristics. Numerical and analytical results show that the maximal EE is achieved by a 'close-to' fully-digital system wherein the number of BS antennas is approximately equal to the number of served small cells.

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“…The condition number is small if these knots are (almost) equidistantly distributed on the unit circle, see [4]. There exist several ideas to overcome these difficulties.…”

Section: Solve the Hankel Systemmentioning

confidence: 99%

Deterministic sparse FFT algorithms

Wannenwetsch

2015

Proc Appl Math and Mech

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In this paper we consider sparse signals ${\bf x}\in {\bf C}^N$ which are known to vanish outside a support interval of length bounded by m < N. For the case that m is known, we propose a deterministic algorithm of complexity ${\cal O}(m\log m)$ for reconstruction of x from its discrete Fourier transform $\widehat{\bf x}\in {\bf C}^N.$ (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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On perfect conditioning of Vandermonde matrices on the unit circle

Cited by 21 publications

References 12 publications

Data Driven Koopman Spectral Analysis in Vandermonde--Cauchy Form via the DFT: Numerical Method and Theoretical Insights

Data Driven Koopman Spectral Analysis in Vandermonde--Cauchy Form via the DFT: Numerical Method and Theoretical Insights

Optimal design of energy-efficient millimeter wave hybrid transceivers for wireless backhaul

Deterministic sparse FFT algorithms

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