Error Formulas for Lagrange Projectors Determined by Cartesian Sets

Li, Zhe; Zhang, Shugong; Dong, Tian; Gong, Yihe

doi:10.1007/s11424-017-6159-8

Cited by 3 publications

(1 citation statement)

References 21 publications

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“…A different application issue is truncation error when using a multivariate interpolating polynomial, even with exact coefficients, to approximate f at a point other than an interpolated node. A remainder formula for any lower set of nodes is shown in [11] and simpler remainder forms, for more structured nodes, Downloaded 07/08/20 to 44.224.250.200. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php are shown in [21] along with a remainder formula for the most general case as in section 6.…”

Section: Discussionmentioning

confidence: 99%

Multivariate Polynomial Interpolation in Newton Forms

Neidinger¹

2019

SIAM Rev.

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Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-difference algorithm for coefficients, generalizes in a straightforward way when interpolating at nodes on a grid within certain regions. While this idea, known as the tensor product case, has been around for over a century, this exposition uses modern multi-index notation to provide generality, concise algorithms, and rigor, appropriate as a topic in introductory numerical analysis. Node configurations where the tensor product case applies are called lower sets, and they include n-dimensional rectangles (boxes) and triangles (corners), the latter having unique interpolation over polynomials of fixed degree. Arbitrary distinct nodes do not ensure unique interpolating polynomials but, when possible, a different basis generalization, which we call Newton--Sauer, results in a nice triangular linear system for the coefficients. For the right number of distinct nodes, an algorithm based on Gaussian elimination produces either this Newton--Sauer basis or a polynomial that is zero on all nodes, showing that unique interpolation is impossible. The algorithm may also be used on any distinct nodes to produce a polynomial subspace of minimal degree where unique interpolation is possible. A variation of this algorithm using matrix block operations produces another basis generalization that we call Newton--Olver, which exactly agrees with the classic one for a corner of nodes and computes an interpolating polynomial using formulas similar to divided differences.

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Section: Discussionmentioning

confidence: 99%

Multivariate Polynomial Interpolation in Newton Forms

Neidinger¹

2019

SIAM Rev.

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The Discrete Approximation Problem for a Special Case of Hermite-Type Polynomial Interpolation

Gong

Jiang²,

Zhang³

2022

J Syst Sci Complex

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Algorithms for computing Gröbner bases of ideal interpolation

Jiang,

Gong

2024

MATH

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<abstract><p>This paper proposes algorithms for computing the reduced Gröbner basis of the vanishing ideal of a finite set of points in the frame of ideal interpolation. We also consider the case that the points have multiplicity conditions. First, we introduce the definition of "reverse" reduced team and compute the interpolation monomial basis of a single point ideal interpolation problem; then we translate the interpolation condition functionals into formal power series via Taylor expansion; this will help convert the general ideal interpolation problem to a single point ideal interpolation problem; and finally, the reduced Gröbner basis is read from formal power series by Gaussian elimination. Our algorithm has a polynomial time complexity, and an example is given to illustrate its effectiveness.</p></abstract>

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Error Formulas for Lagrange Projectors Determined by Cartesian Sets

Cited by 3 publications

References 21 publications

Multivariate Polynomial Interpolation in Newton Forms

Multivariate Polynomial Interpolation in Newton Forms

The Discrete Approximation Problem for a Special Case of Hermite-Type Polynomial Interpolation

Algorithms for computing Gröbner bases of ideal interpolation

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