Bivariate polynomial and continued fraction interpolation over ortho-triples

Wang, Ren-Hong; Qian, Jiang

doi:10.1016/j.amc.2011.02.057

Cited by 5 publications

(7 citation statements)

References 8 publications

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“…then̂( , ) is a general interpolation formula of block based bivariate blending rational interpolation. is block based bivariate Thielelike blending rational interpolation [13]; especially, let = +1; that is to say, every block only includes one point; then,̂( , ) is bivariate Thiele-type branched continued fractions interpolation [11,25]. is block based Newton-Thielelike blending rational interpolation [17]; especially, let = + 1; that is to say, every block only includes one point; then,̂( , ) is bivariate Newton-Thiele blending rational interpolation [11,20].…”

Section: General Interpolation Formulae For Block Based Bivariatementioning

confidence: 99%

“…is block based bivariate Thielelike blending rational interpolation [13]; especially, let = +1; that is to say, every block only includes one point; then,̂( , ) is bivariate Thiele-type branched continued fractions interpolation [11,25]. is block based Newton-Thielelike blending rational interpolation [17]; especially, let = + 1; that is to say, every block only includes one point; then,̂( , ) is bivariate Newton-Thiele blending rational interpolation [11,20]. is block based bivariate associated continued fractions Newton blending rational interpolation [17,20,26]; especially, let = +1; that is to say, every block only includes one point; then, ( , ) is bivariate associated continued fractions Newton blending rational interpolation [17,20,26].…”

Section: General Interpolation Formulae For Block Based Bivariatementioning

confidence: 99%

“…In recent years, Kuchmins'ka and Vozna [6,7], Pahirya [8], Zhao [9], and Wang [10] studied some new kinds of symmetric blending rational interpolation. Wang and Qian studied bivariate polynomial interpolation and continued fractions interpolation over ortho-triples [11]. Zhao and Tan studied block based Newtonlike blending rational interpolation [12] and block based Thiele-like blending rational interpolation [13].…”

Section: Introductionmentioning

confidence: 99%

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A New Approach to General Interpolation Formulae for Bivariate Interpolation

Zou¹,

Tang²

2014

Abstract and Applied Analysis

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General interpolation formulae for bivariate interpolation are established by introducing multiple parameters, which are extensions and improvements of those studied by Tan and Fang. The general interpolation formulae include general interpolation formulae of symmetric branched continued fraction, general interpolation formulae of univariate and bivariate interpolation, univariate block based blending rational interpolation, bivariate block based blending rational interpolation and their dual schemes, and some interpolation form studied by many scholars in recent years. We discuss the interpolation theorem, algorithms, dual interpolation, and special cases and give many kinds of interpolation scheme. Numerical examples are given to show the effectiveness of the method.

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Section: General Interpolation Formulae For Block Based Bivariatementioning

confidence: 99%

Section: General Interpolation Formulae For Block Based Bivariatementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A New Approach to General Interpolation Formulae for Bivariate Interpolation

Zou¹,

Tang²

2014

Abstract and Applied Analysis

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“…Being Inspired by the construction of bivariate polynomial interpolation scheme over ortho-triples in [14], we have considered the case of bivariate continued fraction interpolation over the ortho-triples [18,19], which is based on a new partial inverse divided differences. However, to the best of our knowledge, there are few papers on the study of bivariate interpolation over general triples or nonrectangular mesh.…”

Section: Introductionmentioning

confidence: 99%

Bivariate Polynomial Interpolation over Nonrectangular Meshes

Qian

Zheng

Fan

et al. 2016

Numer. Math. Theory Methods Appl.

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In this paper, bymeans of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

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“…[13][14][15] These results can nevertheless be reduced to a unique irreducible form in the univariate case for the fixed degrees of the numerator and denominator. For multivariate cases, most techniques and algorithms of the univariate case have been generalized, such as the multivariate linear rational interpolation systems, [16][17][18][19] the multivariate continued fraction methods, 10,11,[20][21][22][23][24] and the parameterization of all solutions of the multivariate rational interpolation. 25,26 However, different algorithms yield different results, as we have come to expect, even if in their irreducible forms.…”

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confidence: 99%

The Bulirsch‐Stoer algorithm for multivariate rational interpolation

Xia

Dong

Zhang

et al. 2018

Math Methods in App Sciences

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The Neville‐type algorithms are widely used in engineering and sciences. As an analog of Neville's algorithm that deals with univariate polynomial interpolation, Bulirsch‐Stoer algorithm is a classical one for univariate rational interpolations. It can be applied to calculating the value of the interpolating function at the given point or recovering rational functions. In this paper, we generalize the algorithm to multivariate cases with two versions for different situations. These two generalizations are recursive algorithms. The first one is suitable to calculate the value of the interpolating function and the other one can be applied to recovering multivariate rational functions from accurate measurements. Some two‐variable examples illustrate that, if we recover the rational functions with higher degrees, the second generalization is superior to Thiele‐Thiele continued fraction and two‐variable Löwner matrix methods.

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Bivariate polynomial and continued fraction interpolation over ortho-triples

Cited by 5 publications

References 8 publications

A New Approach to General Interpolation Formulae for Bivariate Interpolation

A New Approach to General Interpolation Formulae for Bivariate Interpolation

Bivariate Polynomial Interpolation over Nonrectangular Meshes

The Bulirsch‐Stoer algorithm for multivariate rational interpolation

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