Third-order accurate monotone cubic Hermite interpolants

Monotonicity-preserving interpolants are used in several applications as engineering or computer aided design. In last years some new techniques have been developed. In particular, in Aràndiga (2013) some new methods to design monotone cubic Hermite interpolants for uniform and non-uniform grids are presented and analyzed. They consist on calculating the derivative values introducing the weighted harmonic mean and a non-linear variation. With these changes, the methods obtained are third-order accurate, except in extreme situations. In this paper, a new general mean is used and a third-order interpolant for all cases is gained. We perform several experiments comparing the known techniques as the method proposed by Fritsch and Butland using the Brodlie’s function, PCHIP program of Matlab (Moler, 2004; Wolberg and Alfy, 2002) with the new algorithm.     

Shape preserving interpolation by cubic G1 splines in $${\mathbb{R}^3}$$

In this paper, G ¹ continuous cubic spline interpolation of data points in , based on a discrete approximation of the strain energy, is studied. Simple geometric conditions on data are presented that guarantee the existence of the interpolant. The interpolating spline is regular, loop-, cusp- and fold-free.      

曲率变化率最小约束下的五次Hermite插值曲线

本文提出了在曲率变化率最小约束条件下的五次Hermite插值曲线算法,与传统的Hermite插值曲线算法相比,利用该算法获得的插值曲线具有更均匀的曲率分布,曲线更光顺,质量更好。     

Dyadic Hermite interpolants on a triangulation

Givenf and f on the vertices of a triangulation, we build an interpolating functionf by means of a subdivision algorithm. Infinite products of matrices are used to prove the convergence to aC ¹ function for some classes of triangulations.     

A degree by degree recursive construction of Hermite spline interpolants

Based on the classical Hermite spline interpolant _H2n−1$H_{2n-1}$, which is the piecewise interpolation polynomial of class Cⁿ⁻¹$C^{n-1}$ and degree 2n−1$2n-1$, a piecewise interpolation polynomial _H2n$H_{2n}$ of degree 2n$2n$ is given. The formulas for computing _H2n$H_{2n}$ by _H2n−1$H_{2n-1}$ and computing _H2n+1$H_{2n+1}$ by _H2n$H_{2n}$ are shown. Thus a simple recursive method for the construction of the piecewise interpolation polynomial set {_Hj}$\left\{H_j\right\}$ is presented. The piecewise interpolation polynomial _H2n$H_{2n}$ satisfies the same interpolation conditions as the interpolant _H2n−1$H_{2n-1}$, and is an optimal approximation of the interpolant _H2n+1$H_{2n+1}$. Some interesting properties are also proved.     

Scalar and Hermite subdivision schemes

A criterion of convergence for stationary nonuniform subdivision schemes is provided. For periodic subdivision schemes, this criterion is optimal and can be applied to Hermite subdivision schemes which are not necessarily interpolatory. For the Merrien family of Hermite subdivision schemes which involve two parameters, we are able to describe explicitly the values of the parameters for which the Hermite subdivision scheme is convergent.     

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and numerical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set,and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the shortcoming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a C1-cubic Hermite interpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global C2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an alternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15(1)(2007), pp. 41-53].     

Hermite四点插指公式

文章利用Hermite插值基函数,将求解Hermite四点插指问题转换为求解8个派生出来的多项式插值问题,证明了Hermite四点插指公式的存在唯一性,并用两种方法构造出Hermite四点插指公式,最后给出了一个算例.     

最简型的Hermite插指

本文提出了Hermite插值问题的一种新形式，幂指数形式，简称Hermite插指。     

Vector cell-average multiresolution based on Hermite interpolation

Harten’s interpolatory multiresolution representation of data has been extended in the case of point-value discretization to include Hermite interpolation by Warming and Beam in [17]. In this work we extend Harten’s framework for multiresolution analysis to the vector case for cell-averaged data, focusing on Hermite interpolatory techniques. *Supported by European Community IHP projects HPRN-CT-2002-00282 and HPRN-CT-2005-00286. **Supported by European Community IHP projects HPRN-CT-2002-00282 and HPRN-CT-2005-00286, and by FPU grant from M.E.C.D. AP2000-1386. ^†Supported by European Community IHP projects HPRN-CT-2002-00282 and HPRN-CT-2005-00286.     

Convergent Vector and Hermite Subdivision Schemes

Hermite subdivision schemes have been studied by Merrien, Dyn, and Levin and they appear to be very different from subdivision schemes analyzed before since the rules depend on the subdivision level. As suggested by Dyn and Levin, it is possible to transform the initial scheme into a uniform stationary vector subdivision scheme which can be handled more easily.With this transformation, the study of convergence of Hermite subdivision schemes is reduced to that of vector stationary subdivision schemes. We propose a first criterion for C⁰-convergence for a large class of vector subdivision schemes. This gives a criterion for C¹-convergence of Hermite subdivision schemes. It can be noticed that these schemes do not have to be interpolatory. We conclude by investigating spectral properties of Hermite schemes and other necessary/sufficient conditions of convergence.     

Error estimates of Hermite interpolation

This paper presents a procedure for obtaining error estimates for Hermite interpolation at the Chebyshev nodes {cos ((2j+1)/2n)} _j ^=0n–1 –1x1, for functionsf(x) of various orders of continuity. The procedure is applicable in many cases when the usual Lagrangian error bound is not, and is a better bound, in general, when both are applicable.     

Piecewise linear interpolants to Lagrange and Hermite convex scattered data

This paper concerns two fundamental interpolants to convex bivariate scattered data. The first,u, is the supremum over all convex Lagrange interpolants and is piecewise linear on a triangulation. The other,l, is the infimum over all convex Hermite interpolants and is piecewise linear on a tessellation. We discuss the existence, uniqueness, and numerical computation ofu andl and the associated triangulation and tessellation. We also describe how to generate convex Hermite data from convex Lagrange data.Research partially supported by the EU Project FAIRSHAPE, CHRX-CT94-0522. The first author was also partially supported by DGICYT PB93-0310 Research Grant.     

On a Hermite interpolation on the sphere

The purpose of this paper is to put forward a kind of Hermite interpolation scheme on the unit sphere. We prove the superposition interpolation process for Hermite interpolation on the sphere and give some examples of interpolation schemes. The numerical examples shows that this method for Hermite interpolation on the sphere is feasible. And this paper can be regarded as an extension and a development of Lagrange interpolation on the sphere since it includes Lagrange interpolation as a particular case.     

Hermite Interpolation and Sobolev Orthogonality

In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g) _S = V(f) A V(g) ^T + <u, f ^(N) g ^(N)V(f) =(f(c ₀), f "(c ₀), ..., f ^{(n – 1)} ₀(c ₀), ..., f(c _p ), f "(c _p ), ..., f ^{(n – 1)} _p(c _p )) u is a regular linear functional on the linear space P of real polynomials, c ₀, c ₁, ..., c _p are distinct real numbers, n ₀, n ₁, ..., n _p are positive integer numbers, N=n ₀+n ₁+...+n _p , and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation.     

About measures and nodal systems for which the Hermite interpolants uniformly converge to continuous functions on the circle and interval

We obtain the Laurent polynomial of Hermite interpolation on the unit circle for nodal systems more general than those formed by the n-roots of complex numbers with modulus one. Under suitable assumptions for the nodal system, that is, when it is constituted by the zeros of para-orthogonal polynomials with respect to appropriate measures or when it satisfies certain properties, we prove the convergence of the polynomial of Hermite-Fejér interpolation for continuous functions. Moreover, we also study the general Hermite interpolation problem on the unit circle and we obtain a sufficient condition on the interpolation conditions for the derivatives, in order to have uniform convergence for continuous functions.Finally, we obtain some improvements on the Hermite interpolation problems on the interval and for the Hermite trigonometric interpolation.     

一类收敛的线性Hermite重心权有理插值

众所周知, Hermite有理插值比Hermite多项式插值具有更好的逼近性, 特别是对于插值点序列较大时, 但很难解决收敛性问题和控制实极点的出现. 本文建立了一类线性Hermite重心有理插值函数$r(x)$,并证明其具有以下优良性质: 第一, 在实数范围内无极点; 第二, 当$k=0,1,2$时,无论插值节点如何分布, 函数$r^{(k)}(x)$具有$O(h^{3d+3-k})$的收敛速度; 第三, 插值函数$r(x)$仅仅线性依赖于插值数据.     

伪二元函数的Hermite插值

将一元函数的Hermite插值方法与伪二元函数结合,得到了伪二元函数的Hermite插值函数,并对插值函数进行了误差分析,最后给出了一个实例.