A Modified Hermite Interpolation with Exponential Parameterization

R. Kozera, M. Wilkołazka

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This work discusses the problem of fitting a regular curve (Formula presented.) based on reduced data points(Formula presented.) in arbitrary Euclidean space. The corresponding interpolation knots (Formula presented.) are assumed to be unknown. In this paper the missing knots are estimated by (Formula presented.) in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter (Formula presented.). In order to fit (Formula presented.), a modified Hermite interpolant (Formula presented.) (a (Formula presented.) piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating (Formula presented.) by (Formula presented.) is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for (Formula presented.) and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all (Formula presented.). A slower linear convergence order in trajectory estimation is established for any (Formula presented.) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data (Formula presented.) and based on (Formula presented.) together with some relevant applications are also briefly recalled in this paper.

Original languageEnglish
Pages (from-to)1-13
Number of pages13
JournalMathematics in Computer Science
DOIs
Publication statusE-pub ahead of print - 6 Jul 2018

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Hermite Interpolation
Parameterization
Interpolation
Sampling
Splines
Trajectories
Industry
Convergence Order
Knot
Interpolate
Shape Preserving
Linear Convergence
Spline Approximation
Sharpness
Mathematica
Arbitrary
Interpolants
Linear Order
Quartic
Hermite

Cite this

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abstract = "This work discusses the problem of fitting a regular curve (Formula presented.) based on reduced data points(Formula presented.) in arbitrary Euclidean space. The corresponding interpolation knots (Formula presented.) are assumed to be unknown. In this paper the missing knots are estimated by (Formula presented.) in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter (Formula presented.). In order to fit (Formula presented.), a modified Hermite interpolant (Formula presented.) (a (Formula presented.) piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating (Formula presented.) by (Formula presented.) is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for (Formula presented.) and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all (Formula presented.). A slower linear convergence order in trajectory estimation is established for any (Formula presented.) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data (Formula presented.) and based on (Formula presented.) together with some relevant applications are also briefly recalled in this paper.",
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A Modified Hermite Interpolation with Exponential Parameterization. / Kozera, R.; Wilkołazka, M.

In: Mathematics in Computer Science, 06.07.2018, p. 1-13.

Research output: Contribution to journalArticle

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