Hermite Interpolation Problem Research Articles

Linear Combination of Independent Exponential Random Variables

In this paper we prove a recursive identity for the cumulative distribution function of a linear combination of independent exponential random variables. The result is then extended to probability density function, expected value of functions of a linear combination of independent exponential random variables, and other functions. Our goal is on the exact and approximate calculation of the above mentioned functions and expected values. We study this computational problem from different views, namely as a Hermite interpolation problem, and as a matrix function evaluation problem. Examples are presented to illustrate the applicability and performance of the methods.

An iterative approach to barycentric rational Hermite interpolation

In this paper we study an iterative approach to the Hermite interpolation problem, which first constructs an interpolant of the function values at $$n+1$$ nodes and then successively adds m correction terms to fit the data up to the mth derivatives. In the case of polynomial interpolation, this simply reproduces the classical Hermite interpolant, but the approach is general enough to be used in other settings. In particular, we focus on the family of rational Floater–Hormann interpolants, which are based on blending local polynomial interpolants of degree d with rational blending functions. For this family, the proposed method results in rational Hermite interpolants, which depend linearly on the data, with numerator and denominator of degree at most $$(m+1)(n+1)-1$$ and $$(m+1)(n-d)$$ , respectively. They converge at the rate of $$O(h^{(m+1)(d+1)})$$ as the mesh size h converges to 0. After deriving the barycentric form of these interpolants, we prove the convergence rate for $$m=1$$ and $$m=2$$ , and show that the approximation results compare favourably with other constructions.

Gibbs–Wilbraham oscillation related to an Hermite interpolation problem on the unit circle

The aim of this piece of work is to study some topics related to an Hermite interpolation problem on the unit circle. We consider as nodal points the zeros of the para-orthogonal polynomials with respect to a measure in the Baxter class and such that the sequence of the first derivative of the reciprocal of the orthogonal polynomials is uniformly bounded on the unit circle. We study the convergence of the Hermite–Fejér interpolants related to piecewise continuous functions and we describe the sets in which the interpolants uniformly converge to the piecewise continuous function as well as the oscillatory behavior of the interpolants near the discontinuities, where a Gibbs–Wilbraham phenomenon appears. Finally we present some numerical experiments applying the main results and by considering nodal systems of interest in the theory of orthogonal polynomials.

On the G2 Hermite Interpolation Problem with clothoids

The G 2 Hermite Interpolation Problem with clothoid curves requires to find the interpolating clothoid that matches initial and final positions, tangents and curvatures, also known as G 2 Hermite data. In the paper we prove that this problem does not always admit solution with only one clothoid segment, nor with two, as some counterexamples show. The general fitting scheme herein proposed requires three arcs determined via the solution of a nonlinear system of 8 equations in 10 unknowns. We discuss how it is possible to recast this system to 2 equations and how to efficiently solve it by means of the Newton method. The choice of the clothoid is crucial because it exhibits the curvature which is linear with the arc length, an important property in many applications ranging from path planning for autonomous vehicles, road design, manufacturing and graphics. The algorithm is tested on a fine hypercube of all possible configurations of angles and curvatures. It always converges and in the worst case it requires 5 standard Newton iterations.

Algebraic-Trigonometric Pythagorean-Hodograph space curves

We introduce a new class of Pythagorean-Hodograph (PH) space curves - called Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) space curves - that are defined over a six-dimensional space mixing algebraic and trigonometric polynomials. After providing a general definition for this new class of curves, their quaternion representation is introduced and the fundamental properties are discussed. Then, as previously done with their quintic polynomial counterpart, a constructive approach to solve the first-order Hermite interpolation problem in ℝ3 is provided. Comparisons with the polynomial case are illustrated to point out the greater flexibility of ATPH curves with respect to polynomial PH curves.

Interpolation mixing hyperbolic functions and polynomials

Exponential polynomials as solutions of differential equations with constant coefficients are widely used for approximation purposes. Recently, mixed spaces containing algebraic, trigonometric and exponential functions have been extensively considered for design purposes. The analysis of these spaces leads to constructions that can be reduced to Hermite interpolation problems. In this paper, we focus on spaces generated by algebraic polynomials, hyperbolic sine and hyperbolic cosine. We present classical interpolation formulae, such as Newton and Aitken-Neville formulae and a suggestion of implementation. We explore another technique, expressing the Hermite interpolant in terms of polynomial interpolants and derive practical error bounds for the hyperbolic interpolant.

Elastic Splines II: Unicity of Optimal s-Curves and Curvature Continuity

Given points $$P_1,P_2,\ldots ,P_m$$ in the complex plane, we are concerned with the problem of finding an interpolating curve with minimal bending energy (i.e., an optimal interpolating curve). It was shown previously that existence is assured if one requires that the pieces of the interpolating curve be s-curves. In the present article, we also impose the restriction that these s-curves have chord angles not exceeding $$\pi /2$$ in magnitude. With this setup, we have identified a sufficient condition for the curvature continuity of optimal interpolating curves. This sufficient condition relates to the stencil angles $$\{\psi _j\}$$ , where $$\psi _j$$ is defined as the angular change in direction from segment $$[P_{j-1},P_j]$$ to segment $$[P_j,P_{j+1}]$$ . An angle $$\Psi $$ ( $$\approx 37^\circ $$ ) is identified, and we show that if the stencil angles satisfy $$\vert {\psi _j} \vert <\Psi $$ , then optimal interpolating curves are curvature continuous. We also prove that the angle $$\Psi $$ is sharp. As with the previous article (Borbely and Johnson in Constr Approx 40:189–218, 2014), much of our effort is concerned with the geometric Hermite interpolation problem of finding an optimal s-curve $$c_1(\alpha ,\beta )$$ that connects $$0+i0$$ to $$1+i0$$ with prescribed chord angles $$(\alpha ,\beta )$$ . Whereas existence was previously shown, and sometimes uniqueness, the present article begins by establishing uniqueness when $$\vert {\alpha } \vert ,\vert {\beta } \vert \le \pi /2$$ and $$\vert {\alpha -\beta } \vert <\pi $$ . We also prove two fundamental identities involving the initial and terminal signed curvatures of $$c_1(\alpha ,\beta )$$ and partial derivatives, with respect to $$\alpha $$ or $$\beta $$ , of the bending energy of $$c_1(\alpha ,\beta )$$ .

The method of Gauss–Newton to compute power series solutions of polynomial homotopies

We consider the extension of the method of Gauss–Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation.

The set of unattainable points for the Rational Hermite Interpolation Problem

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional complete intersection varieties of odd codimension, the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem. Each of these equidimensional varieties can be further decomposed as a union of as many rational (irreducible) varieties as input data points. We exhibit algorithms and equations defining all these objects.

Critical lengths of cycloidal spaces are zeros of Bessel functions

Cycloidal spaces are generated by the trigonometric polynomials of degree one and algebraic polynomials. The critical length of a cycloidal space is the supremum of the lengths of the intervals on which the Hermite interpolation problems are unisolvent. The critical length is related with the critical length for design purposes in computer-aided geometric design. This paper shows an unexpected connection of critical lengths with the zeros of Bessel functions. We prove that the half of the critical length of a cycloidal space is the first positive zero of a Bessel function of the first kind.

Piecewise Chebyshevian splines: interpolation versus design

We consider the wide class of all piecewise Chebyshevian splines with connection matrices at the knots. We prove that a spline space of this class is “good for interpolation” if and only if the spline space obtained by integration is “good for design”. As a consequence, this provides us with a simple practical description of all such spline spaces which can be used for solving Hermite interpolation problems. These results strongly rely on the properties of blossoms.

Hermite interpolation with symmetric polynomials

We study the Hermite interpolation problem on the spaces of symmetric bivariate polynomials. We show that the multipoint Berzolari-Radon sets solve the problem. We also give a Newton formula for the interpolation polynomial and use it to prove a continuity property of the interpolation polynomial with respect to the interpolation points.

Computation of Hermite interpolation in terms of B-spline basis using polar forms

The aim of this paper is to present an Hermite interpolation problem with B-splines of high degree of smoothness. More precisely, we use polar forms to find the B-spline control points for Hermite interpolation. The resulting formula is used to give Hermite basis functions. In particular, Quadratic C1 and cubic C2 interpolations with sharp parameters are analyzed.

Some remarks on cubature formulas with linear operators

In this paper we consider a novel type of cubature formulas called operator-type cubature formulas. The notion originally goes back to a famous work by G. D. Birkhoff in 1906 on Hermite interpolation problem. A well-known theorem by Sobolev in 1962 on invariant cubature formulas is generalized to operator-type cubature, which provides a systematic treatment of Lebedev's works in the 1970s and some related results by Shamsiev in 2006. We give a lower bound for the number of points needed, and discuss analytic conditions for equality, together with tight illustrations for Laplacian-type cubature.

Design with Quasi Extended Chebyshev piecewise spaces

A Quasi Extended Chebyshev (QEC) space is a space of sufficiently differentiable functions in which any Hermite interpolation problem which is not a Taylor problem is unisolvent. On a given interval the class of all spaces which contains constants and for which the space obtained by differentiation is a QEC-space has been identified as the largest class of spaces (under ordinary differentiability assumptions) which can be used for design. As a first step towards determining the largest class of splines for design, we consider a sequence of QEC-spaces on adjacent intervals, all of the same dimension, we join them via connection matrices, so as to maintain both the dimension and the unisolvence. The resulting space is called a Quasi Extended Chebyshev Piecewise (QECP) space. We show that all QECP-spaces are inverse images of two-dimensional Chebyshev spaces under piecewise generalised derivatives associated with systems of piecewise weight functions. We show illustrations proving that QECP-spaces can produce interesting shape effects.

A constructive framework for minimal energy planar curves

Given points P1,P2,…,Pn in the plane, we are concerned with the problem of finding a fair curve which interpolates the points. We assume that we have a method in hand, called a basic curve method, for solving the geometric Hermite interpolation problem of fitting a regular C∞ curve between two points with prescribed tangent directions at the endpoints. We also assume that we have an energy functional which defines the energy of any basic curve. Using this basic curve method repeatedly, we can then construct G1 curves which interpolate the given points P1,P2,…,Pn. The tangent directions at the interpolation points are variable and the idea is to choose them so that the energy of the resulting curve (i.e., the sum of the energies of its pieces) is minimal. We give sufficient conditions on the basic curve method, the energy functional, and the interpolation points for (a) existence, (b) G2 regularity, and (c) uniqueness of minimal energy interpolating curves. We also identify a one-parameter family of basic curve methods, based on parametric cubics, whose minimal energy interpolating curves are unique and G2 under suitable conditions. One member of this family looks very promising and we suggest its use in place of conventional C2 parametric cubic splines.

Arc lengths of rational Pythagorean–hodograph curves

In a recent paper (Lee et al., 2014) a family of rational Pythagorean–hodograph (PH) curves is introduced, characterized by constraints on the coefficients of a truncated Laurent series, and used to solve the first-order Hermite interpolation problem. Contrary to a claim made in this paper, it is shown that these rational PH curves have rational arc length functions only in degenerate cases, where the center of the Laurent series is a real value.

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 \(\mathcal{C}^1\)-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 \(\mathcal{C}^2\)-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].

Shape preserving $$HC^2$$ H C 2 interpolatory subdivision

A subdivision procedure is developed to solve a $$C^2$$ Hermite interpolation problem with the further request of preserving the shape of the initial data. We consider a specific non-stationary and non-uniform variant of the Merrien $$HC^2$$ subdivision family, and we provide a data dependent strategy to select the related parameters which ensures convergence and shape preservation for any set of initial monotone and/or convex data. Each step of the proposed subdivision procedure can be regarded as the midpoint evaluation of an interpolating function—and of its first and second derivatives—in a suitable space of $$C^2$$ functions of dimension $$6$$ which has tension properties. The limit function of the subdivision procedure is a $$C^2$$ piecewise quintic polynomial interpolant.

The Approximation of Hermite Interpolation on the Weighted Mean Norm

We research the simultaneous approximation problem of the higher-order Hermite interpolation based on the zeros of the second Chebyshev polynomials under weighted Lp-norm. The estimation is sharp.