The Vandermonde Matrix

Klinger, Allen

doi:10.2307/2314898

Cited by 53 publications

(26 citation statements)

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“…The inverses of Vandermonde matrices have an analytic form and can be readily determined [11], [12]. Using the results of the current paper, we find an additional approach for inverting Vandermonde matrices.…”

Section: B On Inversion Of Vandermonde Matricesmentioning

confidence: 99%

“…In other words, if is a balanced Vandermonde matrix and is the Fourier transform, then will be a timewarped version of . From (10) it follows that (12) This relationship can be interpreted such that a signal warped, or re-sampled in the time-domain, has the same energy as a signal suitably filtered in the frequency domain. Note that, if our task is to find a convolution matrix for a given time-warping operation , then the diagonal matrix is a free parameter, for which we can choose for example the trivial case,…”

Section: A Euclidean Normsmentioning

confidence: 99%

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Vandermonde Factorization of Toeplitz Matrices and Applications in Filtering and Warping

Bäckström

2013

IEEE Trans. Signal Process.

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By deriving a factorization of Toeplitz matrices into the product of Vandermonde matrices, we demonstrate that the Euclidean norm of a filtered signal is equivalent with the Euclidean norm of the appropriately frequency-warped and scaled signal. In effect, we obtain an equivalence between the energy of frequencywarped and filtered signals. While the result does not provide tools for warping per se, it does show that the energy of the warped signal can be evaluated efficiently, without explicit and complex computation of the warped transform. The main result is closely related to the Vandermonde factorization of Hankel matrices.Application examples for ACELP, perceptual modelling, and time-warped MDCT are discussed.

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Section: B On Inversion Of Vandermonde Matricesmentioning

confidence: 99%

Section: A Euclidean Normsmentioning

confidence: 99%

Vandermonde Factorization of Toeplitz Matrices and Applications in Filtering and Warping

Bäckström

2013

IEEE Trans. Signal Process.

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“…To obtain the relations concerning p h (x; S i,M − ,M + , ∆x) it is not very practical to work with the Newton divided-difference form of p f [15,24], which are widely used in WENO theory [13,12,20,28,29]. It is, instead, preferable to work with the standard form of p f expanded in powers of (x − x i ), whose coefficients can be readily expressed (Proposition 4.5) from the coefficients of the inverse of the Vandermonde matrix [17,27] corresponding to the stencil S i,M − ,M + (Definition 4.1). This representation of p f allows direct use of the formulas relating the coefficients of p h and p f (Lemma 3.1).…”

Section: Polynomial Reconstructionmentioning

confidence: 99%

Approximation error of the Lagrange reconstructing polynomial

Gerolymos

2011

Journal of Approximation Theory

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The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009) 82--126] for the numerical approximation of $f'(x)$ is based on the construction of a dual function $h(x)$ whose sliding averages over the interval $[x-\tfrac{1}{2}\Delta x,x+\tfrac{1}{2}\Delta x]$ are equal to $f(x)$ (assuming an homogeneous grid of cell-size $\Delta x$). We study the deconvolution problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp. Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of $h(x)$ and $f(x)$, and obtain its explicit solution, by introducing rational numbers $\tau_n$ defined by a recurrence relation, or determined by their generating function, $g_\tau(x)$, related with the reconstruction pair of ${\rm e}^x$. We then apply these results to the specific case of Lagrange-interpolation-based polynomial reconstruction, and determine explicitly the approximation error of the Lagrange reconstructing polynomial (whose sliding averages are equal to the Lagrange interpolating polynomial) on an arbitrary stencil defined on a homogeneous grid.Comment: 31 pages, 1 table; revised version to appear in J. Approx. Theor

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“…The definition for π k (X) then follows from the (known) expressions for entries of the inverse of the Vandermonde matrix (cf. [17]). …”

Section: Lemmamentioning

confidence: 99%

Symmetric Functions Capture General Functions

Lipton

Regan

Rudra

2011

Mathematical Foundations of Computer Science 2011

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We show that the set of all functions is equivalent to the set of all symmetric functions up to deterministic time complexity. In particular, for any function f , there is an equivalent symmetric function f sym such that f can be computed form f sym and vice-versa (modulo an extra deterministic linear time computation). For f over finite fields, f sym is (necessarily) over an extension field. This reduction is optimal in size of the extension field. For polynomial functions, the degree of f sym is not optimal. We present another reduction that has optimal degree "blowup" but is worse in the other parameters.

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The Vandermonde Matrix

Cited by 53 publications

References 0 publications

Vandermonde Factorization of Toeplitz Matrices and Applications in Filtering and Warping

Vandermonde Factorization of Toeplitz Matrices and Applications in Filtering and Warping

Approximation error of the Lagrange reconstructing polynomial

Symmetric Functions Capture General Functions

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