Symmetric functions and the Vandermonde matrix

Oruç, Halil; Akmaz, Hakan K.

doi:10.1016/j.cam.2004.01.032

Cited by 25 publications

(29 citation statements)

References 19 publications

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“…In this section we report some results concerning the factorization of a Vandermonde matrix (actually, its transpose), to be used later: though most of them are known (see, for example, [1,9,14,15]), nevertheless, they are here cast in the most general and appropriate form for subsequent reference.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Economical error estimates for block implicit methods for ODEs via deferred correction

Brugnano¹,

Magherini²

2006

Applied Numerical Mathematics

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Deferred correction is a widely used tool for improving the numerical approximation to the solution of ODE problems [10,11,12,13,16,17,18,19,20,21,23]. Indeed, it allows to estimate the error due to the use of discrete methods. Such an estimate may be a global one, in the case of continuous BVPs, or a local one, when IVPs are to be approximated [2,7]. Recently, it has been implemented in the computational code BiM [5] for the numerical solution of stiff ODE-IVPs. In this paper we analyze deferred correction in connection with the methods used in that code, resulting in an overall simplification of the procedure, due to the properties of the underlying methods. The analysis is then extended to more general methods.

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Section: Preliminary Resultsmentioning

confidence: 99%

Economical error estimates for block implicit methods for ODEs via deferred correction

Brugnano¹,

Magherini²

2006

Applied Numerical Mathematics

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“…This clever selection of the nodes reduces the solution of the interpolation problem to the solution of a linear system of order N whose coefficient matrix is the transpose of a nonsingular (since different monomials evaluate to different values) Vandermonde matrix (see, for example, [17] for the expression of the Vandermonde matrix).…”

Section: The Use Of Resultants and The Vandermonde Matrixmentioning

confidence: 99%

Structured matrices in the application of bivariate interpolation to curve implicitization

Marco

Martínez

2007

Mathematics and Computers in Simulation

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A nonstandard application of bivariate polynomial interpolation is discussed: the implicitization of a rational algebraic curve given by its parametric equations. Three different approaches using the same interpolation space are considered, and their respective computational complexities are analyzed. Although the techniques employed are usually associated to numerical analysis, in this case all the computations are carried out using exact rational arithmetic. The power of the Kronecker product of matrices in this application is stressed.

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“…Since the operator ∆ i q1 ∆ j q2 annihilates any polynomial of total degree less than i+j, we see from (4.5) that B m,n (x r y s ; x, y) is a polynomial of total degree min(m + n, r + s). It is shown in [5] that the q-Bernstein polynomial of univariate monomial functions involve the q-Stirling numbers of the second kind S q (k, l), where Note that the q-Stirling numbers of the second kind S q (k, l) also appears in the triangular factorization of the Vandermonde matrix with the geometrically spaced knots, [8]. Another useful representation of (4.7) may be deduced from [11,Chap.…”

Section: Nonparametric Patchesmentioning

confidence: 98%

Tensor product q-Bernstein polynomials

Dişibüyük

Oruç

2008

Bit Numer Math

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An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials Bn,m(f ; x, y) for a function f (x, y), (x, y) ∈ [0, 1] × [0, 1] are defined and fundamental properties are discussed. (2000): 65D17. AMS subject classification

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Symmetric functions and the Vandermonde matrix

Cited by 25 publications

References 19 publications

Economical error estimates for block implicit methods for ODEs via deferred correction

Economical error estimates for block implicit methods for ODEs via deferred correction

Structured matrices in the application of bivariate interpolation to curve implicitization

Tensor product q-Bernstein polynomials

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