Generalized Hermite Interpolation Research Articles

A Two-Dimensional Variant of Newton’s Method and a Three-Point Hermite Interpolation: Fourth- and Eighth-Order Optimal Iterative Schemes

A nonlinear equation f(x)=0 is mathematically transformed to a coupled system of quasi-linear equations in the two-dimensional space. Then, a linearized approximation renders a fractional iterative scheme xn+1=xn−f(xn)/[a+bf(xn)], which requires one evaluation of the given function per iteration. A local convergence analysis is adopted to determine the optimal values of a and b. Moreover, upon combining the fractional iterative scheme to the generalized quadrature methods, the fourth-order optimal iterative schemes are derived. The finite differences based on three data are used to estimate the optimal values of a and b. We recast the Newton iterative method to two types of derivative-free iterative schemes by using the finite difference technique. A three-point generalized Hermite interpolation technique is developed, which includes the weight functions with certain constraints. Inserting the derived interpolation formulas into the triple Newton method, the eighth-order optimal iterative schemes are constructed, of which four evaluations of functions per iteration are required.

Quintic generalized Hermite interpolation curves: construction and shape optimization using an improved GWO algorithm

Quintic generalized Hermite interpolation curves: construction and shape optimization using an improved GWO algorithm

Two-point generalized Hermite interpolation: Double-weight function and functional recursion methods for solving nonlinear equations

Based on the two-point Hermite interpolation technique, the paper proposes a two-point generalized Hermite interpolation and its inversion in terms of weight functions. We prove that upon combining fourth-order optimal iterative scheme to the double Newton’s method (DNM), we can yield a generalized Hermite interpolation formula to relate the first-order derivatives at two points, and the converse is also true. Resorted on the DNM and the derived formula for the generalized inverse Hermite interpolation, some new third-order iterative schemes of quadrature type are constructed. Then, the fourth-order optimal iterative schemes are devised by using a double-weight function. A functional recursion formula is developed which can generate a sequence of two-point generalized Hermite interpolations for any two given weight functions with certain constraints; hence, a more general class of fourth-order optimal iterative schemes is developed from the functional recursion formula. The constructions of fourth-order optimal iterative schemes by using the techniques of double-weight function and the recursion formula obtained from a single weight function are appeared in the literature at the first time. The novelties involve deriving a two-point generalized Hermite interpolation and its inversion in terms of weight functions subjected to two conditions and through the recursion formula, relating the DNM to the third-order iterative schemes by the inverse Hermite interpolation, formulating a functional recursion formula, deriving a broad class fourth-order optimal iterative schemes through double-weight functions rather than the previous technique with a single-weight function, and finding that the new double-weight function and the newly developed fourth-order optimal iterative schemes are inclusive being convergent faster and competitive to other iterative schemes.

A fully discrete pseudospectral method for Fisher's equation on the whole line

In this paper, a fully discrete pseudospectral scheme for Fisher's equation whose solutions behave differently as x→+∞ and x→−∞ is presented using generalized Hermite interpolation. The convergence and the stability of the proposed scheme are analyzed. Numerical results coincide well with theoretical analysis and show efficiency of our algorithm.

On regularity of generalized Hermite interpolation

In this note we study the regularity of generalized Hermite interpolation and compare it to that of classical Hermite interpolation.While every Hermite interpolation scheme is regular in one variable, the “classical Hermite interpolation schemes” in several variables are regular if and only if they are supported at one point. In this note we exhibit some regular generalized Hermite interpolation schemes supported at two points and study some limitation of existence of such schemes. The existence of such schemes provides a class of counterexamples to a conjecture of Jia and Sharma.

Numerical simulation of sound wave propagation using multimoment method of characteristics with general Hermite interpolation

Numerical simulation of sound wave propagation using multimoment method of characteristics with general Hermite interpolation

Padé–type rational and barycentric interpolation

In this paper, we consider the particular case of the general rational Hermite interpolation problem where only the value of the function is interpolated at some points, and where the function and its first derivatives agree at the origin. Thus, the interpolants constructed in this way possess a Pade---type property at 0. Numerical examples show the interest of the procedure. The interpolation procedure can be easily modified to introduce a partial knowledge on the poles and the zeros of the function to approximated. A strategy for removing the spurious poles is explained. A formula for the error is proved in the real case. Applications are given.

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.

A Regularized Solution to the Birkhoff Interpolation Problem

what combination of initial and terminal values suffice to construct a unique solution. This generalizes the concepts of interpolation that were then current. Lagrange interpolation (more properly Warring interpolation, [15, 6]), and the most general Hermite interpolation are special cases if only two time points are considered. See [3] for the most readable account of several different interpolation schemes. The most notable thing about the problem posed by Birkhoff is that there is not always a solution. For example, if n = 2 and we are given u′(0) = 1 and u′(1) = 2 then there is no solution. The most general version of the problem, as posed by Birkhoff, is the following: Let 0 = t0 < t1 < · · · < tN = 1. What combination of n functional values and higher derivatives at the points ti suffices to give a unique solution to the problem. It is this problem that has been studied in the literature for almost one hundred years. Primary references are [12, 8]. Another problem that has been studied and will be shown to be related to Birkhoff interpolation is the study of optimal digital to analog conversion using linear system theory. Two versions of this have been studied, interpolation and smoothing, but

An Improvement to the Computing of Nonlinear Equation Solutions

In this paper, we suggest an improvement to the iterative methods based on the inverse interpolation polynomial, also referred to as the generalized Hermite interpolation, which increases the local order of convergence. A symbolic computation allows us to find the best coefficients with regard to the order of convergence. The adaptation of the strategy presented here gives a new iteration function with a new evaluation of the function. It also shows a smaller cost if we use adaptive multi-precision arithmetic. The numerical results computed using this system, with a floating point system representing 200 and 1000 decimal digits support this theory.

GENERALIZED HERMITE INTERPOLATION AND SAMPLING THEOREM INVOLVING DERIVATIVES

We derive the generalized Hermite interpolation by using the contour integral and extend the generalized Hermite interpolation to obtain the sampling expansion involving derivatives for band-limited functions f, that is, f is an entire function satisfying the following growth condition ｜f(ｚ)｜<TEX>$\leq$</TEX> A exp(<TEX>$\sigma$</TEX>｜y｜) for some A, <TEX>$\sigma$</TEX> > 0 and any ｚ=<TEX>$\varkappa$</TEX> + iy∈C.

Variational Principles and Sobolev-Type Estimates for Generalized Interpolation on a Riemannian Manifold

The purpose of this paper is to study certain variational principles and Sobolev-type estimates for the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation on a closed (i.e., no boundary), compact, connected, orientable, m -dimensional C∞ Riemannian manifold \( \cal M \) , with C∞ metric gij . The rate of approximation can be more fully analyzed with rates of approximation given in terms of Sobolev norms. Estimates on the rate of convergence for generalized Hermite and other distributional interpolants can be obtained in certain circumstances and, finally, the constants appearing in the approximation order inequalities are explicit. Our focus in this paper will be on approximation rates in the cases of the circle, other tori, and the 2 -sphere.

Generalized Hermite Interpolation and Positive Definite Kernels on a Riemannian Manifold

In this paper we introduce a class of positive definite kernels defined on a closed, compact, Riemannian manifold. These kernels provide a grid-free method for solving uniquely a generalized version of the Hermite interpolation problem, a version in which one fits a smooth surface to multi-dimensional scattered data-including data generated by derivatives, fluxes, or any other quantity one can obtain by integrating a function against a compactly supported distribution. The positive definite kernels introduced here include the C∞ members of the class of spherical positive definite functions introduced by Schoenberg [Duke Math, J. 9 (1942), 96-108] and shown to be strictly positive definite by Xu and Cheney [Proc. Amer. Math, Sec.116 (1992), 977-981]. Thus, as a consequence of our results, the generalized Hermite interpolation problem is well-poised-i.e., a solution exists and is unique-for the class of interpolants constructed out of C∞ spherical positive definite functions. We also provide similar results when the underlying manifold is the m-dimensional torus.

Generalized Hermite Interpolation Via Matrix-Valued Conditionally Positive Definite Functions

In this paper, we consider a broad class of interpolation problems, for both scalar- and vector-valued multivariate functions subject to linear side conditions, such as being divergence-free, where the data are generated via integration against compactly supported distributions. We show that, by using certain families of matrix-valued conditionally positive definite functions, such interpolation problems are well poised; that is, the interpolation matrices are invertible. As a sample result, we show that a divergence-free vector field can be interpolated by a linear combination of convolutions of the data-generating distributions with a divergence-free, $3 \times 3$ matrix-valued conditionally positive definite function. In addition, we obtain norm estimates for inverses of interpolation matrices that arise in a class of multivariate Hermite interpolation problems.

Generalized Hermite interpolation via matrix-valued conditionally positive definite functions

In this paper, we consider a broad class of interpolation problems, for both scalar- and vector-valued multivariate functions subject to linear side conditions, such as being divergence-free, where the data are generated via integration against compactly supported distributions. We show that, by using certain families of matrix-valued conditionally positive definite functions, such interpolation problems are well poised; that is, the interpolation matrices are invertible. As a sample result, we show that a divergence-free vector field can be interpolated by a linear combination of convolutions of the data-generating distributions with a divergence-free, $3 \times 3$ matrix-valued conditionally positive definite function. In addition, we obtain norm estimates for inverses of interpolation matrices that arise in a class of multivariate Hermite interpolation problems.

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′2(t)+y′2(t)+z′2(t)≡σ2(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ3 is inherently more complicated than that of the corresponding set in ℝ2. We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bezier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ3), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.

New algorithms for polynomial and trigonometric interpolation on parallel computers

An interpolation polynomial of orderN is constructed fromp independent subpolynomials of ordern ∼ N/p. Each such subpolynomial is found independently and in parallel. Moreover, evaluation of the polynomial at any given point is done independently and in parallel, except for a final step of summation ofp elements. Hence, the algorithm has almost no communication overhead and can be implemented easily on any parallel computer. We give examples of finite-difference interpolation, trigonometric interpolation, and Chebyshev interpolation, and conclude with the general Hermite interpolation problem.

Local generalized Hermite interpolation by quartic C 2 space curves

This paper develops and explains the construction of a piecewise quartic space curve that interpolates positional, tangent, and curvature data. The construction is local and explicit; that is, it does not involve the solution of equations. If only positional data are known, then the tangent and curvature data can he derived by simple local default rules. Another option is to reduce the degree of the curve or minimize the a-norm of its derivative by solving a diagonally dominant, banded, linear system.

Generalizations of Laguerre’s Method: Higher Order Methods

Laguerre’s method is an efficient and reliable method for finding zeros of polynomials and certain other functions. A new derivation and motivation of Laguerre’s method is given, which allows it to be included in a class of methods as general as methods of order three or more based on direct generalized Hermite or hyperosculatory interpolation. Members of this new class share with Laguerre’s method the property of being globally convergent to zeros of polynomials with only real zeros and have the same order of convergence at simple zeros as the classic methods based on generalized Hermite interpolation. Methods of order 4 and 3.303 are investigated and numerical results indicate that for large $(100 \times 100)$ eigenvalue problems the method of order 3.303 is as efficient and reliable as Laguerre’s method.

General lagrange and hermite interpolation in Rn with applications to finite element methods

General lagrange and hermite interpolation in Rn with applications to finite element methods