Conditioning of Rectangular Vandermonde Matrices with Nodes in the Unit Disk

Bazán, Fermín S. V.

doi:10.1137/s0895479898336021

Cited by 57 publications

(56 citation statements)

References 16 publications

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“…By [1,26], the condition number ofW is bounded for large N . ThusW is well conditioned, provided the frequenciesf j (j = 1, .…”

Section: Repeat Step 3 and Solve The Overdetermined Linear Vandermondmentioning

confidence: 99%

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Nonlinear Approximation by Sums of Exponentials and Translates

Peter¹,

Potts²,

Tasche³

2011

SIAM J. Sci. Comput.

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Abstract. In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem. Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands if finitely many perturbed, uniformly sampled data of h are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let ϕ be a given 1-periodic window function as defined in section 4. Further let f be a linear combination of translates of ϕ. Determine all shift parameters, all coefficients, and the number of translates if finitely many perturbed, uniformly sampled data of f are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.

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“…By [1,26], the condition number ofW is bounded for large N . ThusW is well conditioned, provided the frequenciesf j (j = 1, .…”

Section: Repeat Step 3 and Solve The Overdetermined Linear Vandermondmentioning

confidence: 99%

“…. , M), we will be able to recover the original integer M in the case of small error bounds ε and ε 1 .…”

Section: Apm For Sums Of Translatesmentioning

confidence: 99%

Nonlinear Approximation by Sums of Exponentials and Translates

Peter¹,

Potts²,

Tasche³

2011

SIAM J. Sci. Comput.

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show abstract

“…The equalities (18) follow on analyzing the eigenvalues of C A (m, 1)C * A (m, 1) from this equality; details can be found in [5].…”

Section: Theorem 23 Let the Singular Values Of C A (M Q) Be Arrangmentioning

confidence: 99%

“…The paper is organized as follows. In Section 2, we describe results concerning the singular values of projected companion matrices by extending the work in [5]. The results obtained are then exploited in Section 3, in which we analyze the departure of the projected companion matrix from normality.…”

Section: Introductionmentioning

confidence: 99%

Matrix polynomials with partially prescribed eigenstructure: eigenvalue sensitivity and condition estimation

Bazán¹

2005

Mat. apl. comput.

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Abstract. Let P m (z) be a matrix polynomial of degree m whose coefficients A t ∈ C q×q satisfy a recurrence relation of the form:and L ∈ C n×q . The coefficients are not uniquely determined from the recurrence relation but the polynomials are always guaranteed to have n fixed eigenpairs, {z j , l j }, where l j is the jth column of L * . In this paper, we show that the z j 's are also the n eigenvalues of an n × n matrix C A ; based on this result the sensitivity of the z j 's is investigated and bounds for their condition numbers are provided. The main result is that the z j 's become relatively insensitive to perturbations in C A provided that the polynomial degree is large enough, the number n is small, and the eigenvalues are close to the unit circle but not extremely close to each other. Numerical results corresponding to a matrix polynomial arising from an application in system theory show that low sensitivity is possible even if the spectrum presents clustered eigenvalues.Mathematical subject classification: 65F20,65F15.

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“…Reference [7] derives bounds on the condition number (through bounds on the singular values) of Vandermonde matrices with nodes on the unit circle, dependant on the minimum and maximum distances between the nodes of the generating row. Another reference( [1]) considered the effect of increasing N on the conditioning of n × N rectangular Vandermonde matrices with nodes in the unit disk.…”

Section: Introduction and Some Notationmentioning

confidence: 99%

On perfect conditioning of Vandermonde matrices on the unit circle

Berman¹,

Feuer²

2007

ELA

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Abstract. Let K, M ∈ N with K < M, and define a square K × K Vandermonde matrix A = A τ, − → n ¡ with nodes on the unit circle: Ap,q = exp (−j2πpnqτ /K) ; p, q = 0, 1, ..., K − 1, where nq ∈ {0, 1, ..., M − 1} and n 0 < n 1 < .... < n K−1 . Such matrices arise in some types of interpolation problems. In this paper, necessary and sufficient conditions are presented on the vector − → n so that a value of τ ∈ R can be found to achieve perfect conditioning of A. A simple test to check the condition is derived and the corresponding value of τ is found.

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Conditioning of Rectangular Vandermonde Matrices with Nodes in the Unit Disk

Cited by 57 publications

References 16 publications

Nonlinear Approximation by Sums of Exponentials and Translates

Nonlinear Approximation by Sums of Exponentials and Translates

Matrix polynomials with partially prescribed eigenstructure: eigenvalue sensitivity and condition estimation

On perfect conditioning of Vandermonde matrices on the unit circle

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