Showing papers on "Monotone cubic interpolation published in 1995"
TL;DR: Two efficient algorithms for enclosing a zero of a continuous function that make essential use of inverse cubic interpolation are presented and it is proved that the second algorithm is optimal in a certain family.
Abstract: Two efficient algorithms for enclosing a zero of a continuous function are presented. They are similar to the recent methods, but together with quadratic interpolation they make essential use of inverse cubic interpolation as well. Since asymptotically the inverse cubic interpolation is always chosen by the algorithms, they achieve higher-efficiency indices: 1.6529… for the first algorithm, and 1.6686… for the second one. It is proved that the second algorithm is optimal in a certain family. Numerical experiments show that the two new methods compare well with recent methods, as well as with the efficient solvers of Dekker, Brent, Bus and Dekker, and Le. The second method from the present article has the best behavior of all 12 methods especially when the termination tolerance is small.
72 citations
TL;DR: In this paper, the performance of standard cubic Hermite interpolation can be improved by interpolating a third point within the parameter interval, and the resulting method is easy to implement and achieves the optimal approximation order 5.
53 citations
TL;DR: In this paper, a general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives, and an interpolation formula analogous to that of Newton in one variable is established.
Abstract: We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.
48 citations
TL;DR: It is shown that the E(3) cubic spline is more accurate than the second one, and also, it has superconvergence properties which the not-a-knot cubic splines does not have.
22 citations
TL;DR: Sharp error bounds for interpolating splines in tension with variable tension parameters are considered and error bounds sharper than those previously published are developed for derivatives.
21 citations
TL;DR: A Lagrangian parameter approach to problems of best constrained approximation in Hilbert space is reviewed, which is applied to the problem of interpolation of data in a plane by a cubic spline function which is subject to obstacles.
Abstract: We review a Lagrangian parameter approach to problems of best constrained approximation in Hilbert space. The variable is confined to a closed convex subset of the Hilbert space and is also assumed to satisfy linear equalities. The technique is applied to the problem of interpolation of data in a plane by a cubic spline function which is subject to obstacles. The obstacles may be piecewise cubic polynomials over the original knot set. A characterization result is obtained which is used to develop a Newton-type algorithm for the numerical solution.
12 citations
TL;DR: The Hermite interpolation problem by bivariate algebraic polynomials is investigated in this paper, where the interpolation parameters are the values of a function and its partial derivatives up to some ordern at the nodesz and the space of the polynomial is π n (R 2 ) := polynoms of total degree≤n.
Abstract: The Hermite interpolation problem by bivariate algebraic polynomials is investigated. The interpolation parameters are the values of a function and its partial derivatives up to some ordern at the nodesz and the space of the polynomials is π
n
(R
2) := polynomials of total degree≤n. We describe the class of almost-regular Hermite interpolation schemes under some restrictions and give a complete description of perfect singular schemes.
12 citations
TL;DR: It is shown that if the chosen partitioning of the state space is sufficiently small, the resulting suboptimal controller leads to a stable closed loop system.
Abstract: We consider a class of nonlinear quadratic
regulator problems where the system dynamics are affine in the control.
It has been shown recently that an optimal feedback control law
for this class of problems can be given in terms of the solution of a
state dependent algebraic Ricatti equation (ARE) at each instance of
time. However, in most practical
problems it is not possible to find an analytic solution to the ARE and hence
numerical schemes to calculate suboptimal controls are required. In this paper,
we consider one such scheme based on cubic basis spline interpolation. It is
shown that if the chosen partitioning of the state space is sufficiently small,
the resulting suboptimal controller leads to a stable closed loop system.
11 citations
TL;DR: This paper considers the optimality and the evaluation of the constants that appear in the expressions of error bounds for interpotating spline functions over a uniform mesh of the real line when the nodes are uniformly shifted.
TL;DR: Two quite universal methods for constructing generalized B splines are proposed based on solving directly a system and on the formulae obtained are shown to form weak Chebyshev systems.
Abstract: Direct and recursive algorithms are proposed for constructing generalized cubic B splines Explicit formulae are obtained for generalized B splines Properties of series con sisting of B splines are studied It is shown that generalized B splines form weak Chebyshev systems The presented formulae of local approximation are exact for polynomials of the rst degree Examples of generalized B splines including those with alternating signs are considered The tool of generalized cubic splines is widely used in solving problems of isogeometric interpolation Introducing one or another of parameters into the spline structure we can preserve such characteristics of the initial data as the convexity monotonicity linear and plane pieces etc Here the main problem is to develop an algorithm for choosing parameters automatically The available algorithms are based mainly on the piecewise representation of splines The method of local optimization combined with recursive algo rithms for calculating polynomial B splines was found to be e cient in practical applications Although the theory of generalized B splines is well de veloped they are not applied widely for solving problems of isogeometric approximation This is due to fact that there are no simple and e cient com putational algorithms and explicit formulae for generalized B splines They are developed only for trigonometric hyperbolic and some special kinds of more general Chebyshev splines which have a limited eld of application In this paper we propose two quite universal methods for constructing generalized B splines The rst method is based on solving directly a system y Institute of Computational Technologies Russian Academy of Sciences Novosibirsk Russia
TL;DR: In this article, the application of cubic Hermitian interpolation based finite element schemes for the time integration of the differential-algebraic system arising in the dual reciprocity boundary element formulation of transient diffusion problems is presented.
Abstract: This paper presents the application of cubic Hermitian interpolation based finite element schemes for the time integration of the differential-algebraic system arising in the dual reciprocity boundary element formulation of transient diffusion problems. Weighted residual procedure is used to obtain the desired recurrence relations. Numerical results presented for three representative problems involving different types of boundary conditions amply demonstrate the high accuracy of the cubic Hermitian schemes.
TL;DR: In this article, a Hermite interpolant for regular simplices is presented, which is unique under the given conditions and has a polynomial degree of only 3, but is the identity for quadratic polynomials.
01 Jan 1995
TL;DR: In a recent lecture at the NATO Advanced Studies Institute as discussed by the authors, the authors tried to acquaint the audience with the very recent progress that has been made in the approximation of functions defined on spheres.
Abstract: The purpose of this lecture at the NATO Advanced Studies Institute was purely expository. I tried to acquaint the audience with the very recent progress that has been made in the approximation of functions defined on spheres. For practical purposes, the sphere of greatest interest would appear to be S2, since it serves as a (flawed) model of the Earth’s surface. There is every reason to believe that the new approximation methods now being developed will be found useful in geological exploration and meteorological modelling, to name just two areas of application.
TL;DR: In this paper, it was proved that the knots are to be at the midpoints of every two adjacent adjacent data points for the smoothest interpolation of equispaced data points.
01 Jan 1995
TL;DR: In this article, the problem of Hermite interpolation on scattered data using ridge functions was studied, and a wide class of ridge functions that may be used in practice to interpolate Hermite data was presented.
Abstract: In this paper we study the problem of Hermite interpolation on scattered data using ridge functions, and give a wide class of ridge functions that may be used in practice to interpolate Hermite data.
01 Jan 1995
TL;DR: In this article, a variational method of regularization was proposed to solve nonlinear ill-posed problems involving monotone operators in infinite dimensional Banach space, when perturbative operators are non-monotone, basing on minimization of norm in interpolation space over closed and convex sets.
Abstract: In this note we study a variational method of regularization to solve nonlinear ill-posed problems involving monotone operators in infinite dimensional Banach space, when perturbative operators are non - monotone, basing on minimization of norm in interpolation space over closed and convex sets.
Journal Article•
TL;DR: A new method-the method of B-spline generatix is proposed, by which interpolation surfaces of C2 continuity can be designed, and, the surface design in mLinufacturing such as die-machining t ship-building, etc.
Abstract: Maily popular design methods of free-form surface in CAD/CAM, such as Bsplins surface, Bener surface, Ball stlrface, etc, only can be used to clesigll fitting, not interpolation surface, and are not fit for designing interpolation surfaces in engneering. In this work, a new method-the method of B-spline generatix is proposed, by which interpolation surfaces of C2 continuity can be designed. And, the surface design in mLinufacturing such sa die-machining t ship-building, etc.
01 Nov 1995
TL;DR: Considering a given function f ∈ C 4 and its unique deficient cubic spline interpolant, which matched the given function and its derivative at mid point between the successive mesh point, this paper obtained an asymptotically precise estimate fors′ -f′.
Abstract: Considering a given functionf ∈C 4 and its unique deficient cubic spline interpolant, which match the given function and its derivative at mid point between the successive mesh point, we have obtained in the present paper asymptotically precise estimate fors′ -f′.
TL;DR: In this article, the applicability of cubic pseudosplines is explored via numerical comparison with previously obtained results from the implementation of the quintic ones for ray-optical field evaluations, when the antenna secondary field is calculated by integration of physical-optics-induced currents.
Abstract: A recent study demonstrated the superior stability of quintic pseudosplines, when compared with usual oscillatory series expansions, for global interpolation of numerically defined reflector surfaces presenting critical behavior of their principal curvatures. Although interpolating functions with continuous derivatives up to second order are necessary for ray-optical field evaluations, C-class surface descriptions are sufficient when the antenna secondary field is calculated by integration of physical-optics-induced currents. In this connection, the applicability of cubic pseudosplines is explored via numerical comparison with previously obtained results from the implementation of the quintic ones. © 1995 John Wiley & Sons. Inc.
TL;DR: Rational curves are a useful class of curves for CAGD and CAM by showing that a BR-cubic is the projective transform of a cubic Bezier and emphasizing some geometrical properties of a rational cubic.
TL;DR: In this paper, two phase unwrapping criteria appropriate to different conditions for voiced speech synthesis are suggested, namely, when the synthesis frame lags the analysis frame by half a frame, the first one, named Criterion A, should be used, which leads to the same unwrapings results when used for both quadratic and cubic interpolation.
Abstract: In this paper, two-phase unwrapping criteria appropriate to different conditions for voiced speech synthesis are suggested. When the synthesis frame lags the analysis frame by half a frame, the first one, named Criterion A, should be used. It leads to the same unwrapping results when it is used for both quadratic and cubic interpolation. In addition, for quadratic interpolation, Criterion A is equivalent to the criterion to minimize the absolute frequency deviation, suggested by Griffin and Lim (1988); for cubic interpolation, it is equivalent to the 'maximally smooth’ criterion, proposed by McAulay and Quatieri (1985). However, when the synthesis frame coincides with the analysis frame, the second criterion, named Criterion B, should be used. It also leads to the same unwrapping results for both interpolation methods. In the second case, simulations using the multi-band excitation (MBE) model to synthesize speech have shown that Criterion B outperforms Criterion A.
27 Nov 1995
TL;DR: The solution presented falls into two parts, the first being the formulation of an abstract simplex mesh that parametrizes the data while the second part presents an interpolation algorithm based on the structure of a simple feed-forward perceptron.
Abstract: In this paper we present a solution to a C/sup 1/ interpolation problem for lower dimensional data in Euclidean space. The solution presented falls into two parts, the first being the formulation of an abstract simplex mesh that parametrizes the data while the second part presents an interpolation algorithm based on the structure of a simple feed-forward perceptron. To emphasise the connection between our approach and classical spline interpolation we choose cubic polynomial activation functions in the neural units of the perceptron.