Divided Differences and Two-sided Polynomial Interpolation Over Quaternions

2017 ◽  
pp. 19-35
Author(s):  
Vladimir Bolotnikov
Keyword(s):  
Polynomial Interpolation ◽  
Divided Differences

scholarly journals A New Approach to Newton-Type Polynomial Interpolation with Parameters

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Le Zou ◽  
Liangtu Song ◽  
Xiaofeng Wang ◽  
Thomas Weise ◽  
Yanping Chen ◽  
...  
Keyword(s):  
Polynomial Interpolation ◽  
Information Matrix ◽  
Divided Differences ◽  
Mathematical Representation ◽  
Interpolation Algorithm ◽  
New Approach ◽  
Multiple Parameters ◽  
Parameter Values ◽  
The Given ◽  
Interpolation Functions

Newton’s interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. The interpolation function of most classical approaches is unique to the given data. In this paper, univariate and bivariate parameterized Newton-type polynomial interpolation methods are introduced. In order to express the divided differences tables neatly, the multiplicity of the points can be adjusted by introducing new parameters. Our new polynomial interpolation can be constructed only based on divided differences with one or multiple parameters which satisfy the interpolation conditions. We discuss the interpolation algorithm, theorem, dual interpolation, and information matrix algorithm. Since the proposed novel interpolation functions are parametric, they are not unique to the interpolation data. Therefore, its value in the interpolant region can be adjusted under unaltered interpolant data through the parameter values. Our parameterized Newton-type polynomial interpolating functions have a simple and explicit mathematical representation, and the proposed algorithms are simple and easy to calculate. Various numerical examples are given to demonstrate the efficiency of our method.

scholarly journals On the divided differences of the remainder in polynomial interpolation

2004 ◽  
Vol 127 (2) ◽  
pp. 193-197 ◽  
Author(s):  
Xinghua Wang ◽  
Ming-Jun Lai ◽  
Shijun Yang
Keyword(s):  
Polynomial Interpolation ◽  
Divided Differences

On polynomial interpolation at the points of a geometric progression

1981 ◽  
Vol 90 (3-4) ◽  
pp. 195-207 ◽  
Author(s):  
I. J. Schoenberg
Keyword(s):  
Quadrature Formula ◽  
Polynomial Interpolation ◽  
Theta Functions ◽  
Important Case ◽  
Divided Differences ◽  
Geometric Progression ◽  
Close Connection ◽  
Numerical Examples ◽  
Interpolation Series

SynopsisThis note pursues two aims: the first is historical and the second is factual.1. We present James Stirling's discovery (1730) that Newton's general interpolation series with divided differences simplifies if the points of interpolation form a geometric progression. For its most important case of extrapolation at the origin. Karl Schellbach (1864) develops his algorithm of q-differences that also leads naturally to theta-functions. Carl Runge (1891) solves the same extrapolation at the origin, without referring to the Stirling-Schellbach algorithm. Instead, Runge uses “Richardson's deferred approach to the limit” 20 years before Richardson.2. Recently, the author found a close connection to Romberg's quadrature formula in terms of “binary” trapezoidal sums. It is shown that the problems of Stirling, Schellbach, and Runge, are elegantly solved by Romberg's algorithm. Numerical examples are given briefly. Fuller numerical details can be found in the author's MRC T.S. Report #2173, December 1980, Madison, Wisconsin. Thanks are due to the referee for suggesting the present stream-lined version.

Bivariate Polynomial Interpolation over Nonrectangular Meshes

2016 ◽  
Vol 9 (4) ◽  
pp. 549-578
Author(s):  
Jiang Qian ◽  
Sujuan Zheng ◽  
Fan Wang ◽  
Zhuojia Fu
Keyword(s):  
Tensor Product ◽  
Polynomial Interpolation ◽  
Recursive Algorithm ◽  
Scattered Data ◽  
High Order ◽  
Divided Differences ◽  
Bivariate Polynomials ◽  
Bivariate Polynomial ◽  
Bivariate Polynomial Interpolation ◽  
Interpolating Polynomials

AbstractIn 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.

scholarly journals Profiled Horn Antenna with Wideband Capability Targeting Sub-THz Applications

Electronics ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 412
Author(s):  
Jay Gupta ◽  
Dhaval Pujara ◽  
Jorge Teniente
Keyword(s):  
Hermite Polynomial ◽  
Polynomial Interpolation ◽  
Horn Antenna ◽  
Submillimeter Wavelengths ◽  
Thz Applications ◽  
Millimeter And Submillimeter Wavelengths ◽  
Better Than

This paper proposes a wideband profiled horn antenna designed using the piecewise biarc Hermite polynomial interpolation and validated experimentally at 55 GHz. The proposed design proves S11 and directivity better than −22 dB and 25.5 dB across the entire band and only needs 3 node points if compared with the well-known spline profiled horn antenna. Our design makes use of an increasing radius and hence does not present non-accessible regions from the aperture, allowing its fabrication with electro erosion techniques especially suitable for millimeter and submillimeter wavelengths.

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