Multivariate Polynomial Interpolation

Gasca, M.

doi:10.1007/978-94-009-2017-0_7

Cited by 25 publications

(12 citation statements)

References 11 publications

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“…Although Gasca [Gas90] observes that there is some freedom, his construction does not clarify the role of freedom in choosing the lines and points. For our purposes, it is essential to explore the nature of this non-uniqueness in order to specify certain interesting point-line configurations associated with Newton L-bases.…”

Section: Newton Basesmentioning

confidence: 92%

“…Thus we can associate a total of ( n+2 2 ) points counted with appropriate multiplicity to a Newton L-basis defined by 2n lines. This point-line configuration associated with a Newton L-basis is a subclass of the point-line configurations that form the starting point for the construction of Newton bases defined by Gasca [Gas90]. With this associated point-line configuration, it can be readily verified that the Newton L-bases defined here can be realized as special cases of the bivariate Newton bases defined by Gasca.…”

Section: Newton Basesmentioning

confidence: 93%

“…To justify the terminology Newton basis, we are going to establish that these Newton L-bases are special cases of the bivariate Newton bases defined by Gasca [Gas90]. Gasca starts with a particular set of lines and points and associates a Newton basis to this point-line configuration.…”

Section: Newton Basesmentioning

confidence: 99%

“…In fact, the coefficients of the solution can be interpreted as the generalization of divided differences to higher dimensions. Further discussion of the extremely important role the Newton bases play in multivariate interpolation and approximation can be found in [Gas90].…”

Section: Newton Basesmentioning

confidence: 99%

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Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Lodha¹,

Goldman²

1998

Journal of Approximation Theory

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L-Bases and B-bases are two important classes of polynomial bases used for representing surfaces in approximation theory and computer aided geometric design. It is well known that the Bernstein and multinomial (or Taylor) bases are special cases of both L-bases and B-bases. We establish that certain proper subclasses of bivariate Lagrange and Newton bases are L-bases. Furthermore, we present a rich collection of lattices (or point-line configurations) that admit unique Lagrange or Hermite interpolation problems which can be solved quite naturally in terms of Lagrange and Newton L-bases. A new geometric point-line duality between L-bases and B-bases is described: lines in L-bases correspond to points or vectors in B-bases and concurrent lines map to collinear points and vice versa. This duality between L-bases and B-bases is then used to establish that certain proper subclasses of power bases are B-bases and are dual to Lagrange L-bases. This geometric duality is further used to describe the lattices that admit power B-bases. B-bases dual to Newton L-bases are also investigated. Duality can also be used to develop change of basis algorithms with computational complexity O(n 3 ) between any two L-bases andÂor B-bases. We describe, in particular, a new change of basis algorithm from a bivariate Lagrange L-basis to a bivariate Bernstein basis with computational complexity O(n 3 ).

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Section: Newton Basesmentioning

confidence: 92%

Section: Newton Basesmentioning

confidence: 93%

Section: Newton Basesmentioning

confidence: 99%

Section: Newton Basesmentioning

confidence: 99%

See 2 more Smart Citations

Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Lodha¹,

Goldman²

1998

Journal of Approximation Theory

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“…However, such conditions are usually too restrictive and difficult to fulfill and apply. After all, the set of nodes for which Lagrange interpolation is not unique has measure zero, and interpolation is almost always possible; for literature and a historical account we refer to the recent survey [2] and the monograph [4], the latter one also containing a particularly extensive bibliography. Recently, a very interesting approach has been given by de Boor and Ron (see [1] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

On Multivariate Lagrange Interpolation

Sauer¹,

Xu²

1995

Mathematics of Computation

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Abstract. Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formula for Lagrange interpolation of degree « of a function /, which is a sum of integrals of certain (n + l)st directional derivatives of / multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.

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A Case Study in Multivariate Lagrange Interpolation

Sauer

1995

Approximation Theory, Wavelets and Applications

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Multivariate Polynomial Interpolation

Cited by 25 publications

References 11 publications

Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

On Multivariate Lagrange Interpolation

A Case Study in Multivariate Lagrange Interpolation

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