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Showing papers on "Monotone cubic interpolation published in 2013"


Journal ArticleDOI
TL;DR: In this article, the meshless local radial point interpolation (MLRPI) method is applied to simulate a nonlinear partial integro-differential equation arising in population dynamics.
Abstract: In this paper the meshless local radial point interpolation (MLRPI) method is applied to simulate a nonlinear partial integro-differential equation arising in population dynamics. This PDE is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. In MLRPI method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method is proposed to construct shape functions using the radial basis functions. A one-step time discretization method is employed to approximate the time derivative. To treat the nonlinearity, a simple predictor–corrector scheme is performed. Also the integral term, which is a kind of convolution, is treated by the cubic spline interpolation. The numerical studies on sensitivity analysis and convergence analysis show that our approach is stable. Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method.

83 citations


Journal ArticleDOI
TL;DR: In this article, a simple explicit construction for a Open image in new window-cubic Hermite Fractal Interpolation Function (FIF) under some suitable hypotheses on the original function was established.
Abstract: The theory of splines is a well studied topic, but the kinship of splines with fractals is novel We introduce a simple explicit construction for a Open image in new window-cubic Hermite Fractal Interpolation Function (FIF) Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the Lp-norm, 1≤p≤∞) for the interpolation error of the Open image in new window-cubic Hermite FIF and its first derivative Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys Open image in new window global smoothness Consequently, our method offers an alternative to the standard moment construction of Open image in new window-cubic spline FIFs Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting Open image in new window-cubic FIF lies within a prescribed rectangle These parameters include, in particular, conditions for the positivity of the cubic FIF Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials We also provide numerical examples to corroborate our results

81 citations


Journal ArticleDOI
TL;DR: It is shown that the interpolation problem for multiple knot cardinal splines subject to general interpolation conditions has a unique solution with polynomial growth if the data grow correspondingly provided a certain determinantal condition is satisfied.
Abstract: In the present paper it is shown that the interpolation problem for multiple knot cardinal splines subject to general interpolation conditions has a unique solution with polynomial growth if the data grow correspondingly provided a certain determinantal condition is satisfied. An application to Hs error estimates for the interpolation with periodic multiple knot splines is given. 2010 Mathematical subject classification: 65M12

56 citations


Journal ArticleDOI
TL;DR: The image, preimage, and cartesian product of cubic K U-ideals of KU-algebras are defined and several results are presented.
Abstract: We introduce the notion of cubic KU-ideals of KU-algebras and several results are presented in this regard. The image, preimage, and cartesian product of cubic KU-ideals of KU-algebras are defined.

45 citations


Journal ArticleDOI
TL;DR: In this paper, a piecewise rational function in cubic/quadratic form involving three shape parameters is presented to preserve the inherited shape feature (positivity) of data and the remaining two shape parameters are left free for the designer to modify the shape of positive curves as per industrial needs.
Abstract: This work addresses the shape preserving interpolation problem for visualization of positive data. A piecewise rational function in cubic/quadratic form involving three shape parameters is presented. Simple data dependent conditions for a single shape parameter are derived to preserve the inherited shape feature (positivity) of data. The remaining two shape parameters are left free for the designer to modify the shape of positive curves as per industrial needs. The interpolant is not only C, local, computationally economical, but it is also a visually pleasant and smooth in comparison with existing schemes. Several numerical examples are supplied to illustrate the proposed interpolant.

28 citations


Journal ArticleDOI
TL;DR: In this article, a new 9 point compact discretization of order two in y -and order four in x -directions, based on cubic spline approximation, for the solution of two dimensional quasi-linear elliptic partial differential equations is reported.

26 citations


Book
18 Oct 2013
TL;DR: The exponential spline as discussed by the authors is a generalization of the semiclassical cubic spline known in the literature as the exponential splines, which can preserve convexity and monotonicity present in the data.
Abstract: Herein, we discuss a generalization of the semiclassical cubic spline known in the literature as the exponential spline In actuality, the exponential spline represents a continuum of interpolants ranging from the cubic spline to the linear spline A particular member of this family is uniquely specified by the choice of certain "tension" parameters We first outline the theoretical underpinnings of the exponential spline This development roughly parallels the existing theory for cubic splines The primary extension lies in the ability of the exponential spline to preserve convexity and monotonicity present in the data We next discuss the numerical computation of the exponential spline A variety of numerical devices are employed to produce a stable and robust algorithm An algorithm for the selection of tension parameters that will produce a shape preserving approximant is developed A sequence of selected curve-fitting examples are presented which clearly demonstrate the advantages of exponential splines over cubic splines We conclude with a consideration of the broad spectrum of possible uses of exponential splines in the applications Our primary emphasis is on computational fluid dynamics although the imaginative reader will recognize the wider generality of the techniques developed

24 citations


Journal ArticleDOI
TL;DR: A new approach to rational spline motion design is described by using techniques of geometric interpolation which enables us to reduce the discrepancy in the number of degrees of freedom of the trajectory of the origin and of the rotational part of the motion.

16 citations


Journal ArticleDOI
TL;DR: Based on cubic Hermite interpolation, a two-level method is presented for the numerical solutions of one-dimensional telegraph equations with Dirichlet boundary conditions and it is proved that the scheme is unconditionally stable.

16 citations


Journal ArticleDOI
TL;DR: This work rigorously proves that the PH interpolant it selects doesn’t depend on the unit pure vector chosen for representing its hodograph in quaternion form, and evaluates the corresponding interpolation scheme from a theoretical point of view, proving with the help of symbolic computation that it has fourth approximation order.

14 citations


Journal ArticleDOI
02 Dec 2013
TL;DR: This paper presents a scalable edge map to recover high frequency components of edge regions in up-scaled images to improve the sharpness and use a range compression method to reduce ringing artifacts.
Abstract: In this paper, we propose an edge map up-scaling method. We propose an edge curve scaling method with cubic spline interpolation to up-scale an edge map. If an edge curve is directly applied to the cubic spline interpolation function for edge curve up-scaling, the edge curve scaling results have zigzag artifacts. We also propose a simple smoothing function to avoid the zigzag problems and maintain the contour shape of images. By predicting edge regions of the up-scaled image, we can recover high frequency components of edge regions of the up-scaled image to improve the sharpness and reduce ringing artifacts.

Journal ArticleDOI
TL;DR: This paper presents and analyze two new methods to prescribe the derivatives at the breakpoints that lead to third order approximations even in the case of nonuniform grids, and demonstrates that with these alternatives the monotonicity may only be lost in rather extreme situations.
Abstract: Monotonicity-preserving approximation methods are used in numerous applications. With them we can reconstruct a function from a discrete set of data while preserving its monotonicity properties. In this paper we analyze monotone piecewise cubic Hermite interpolants for uniform and nonuniform grids. We present and analyze two new methods to prescribe the derivatives at the breakpoints that lead to third order approximations even in the case of nonuniform grids. We demonstrate that with these alternatives the monotonicity may only be lost in rather extreme situations. In such cases we propose modifications of the algorithms that guarantee the monotonicity but with local second order accuracy. We also perform several numerical experiments which exemplify the properties of the proposed algorithms and compare them with the technique proposed by Fritsch and Butland, in which the derivatives at the breakpoints are calculated using Brodlie's function. It is available in Netlib (PCHIP.FOR) and it is also used in t...

Journal ArticleDOI
TL;DR: Hermite interpolation using polynomial splines on T-meshes is discussed in detail, leading to an error bound for interpolation of smooth functions.

Journal ArticleDOI
TL;DR: Numerical tests indicate that the proposed piecewise bivariate Hermite interpolations are economic and have good approximation capacity.
Abstract: The requirements for interpolation of scattered data are high accuracy and high efficiency. In this paper, a piecewise bivariate Hermite interpolant satisfying these requirements is proposed. We firstly construct a triangulation mesh using the given scattered point set. Based on this mesh, the computational point () is divided into two types: interior point and exterior point. The value of Hermite interpolation polynomial on a triangle will be used as the approximate value if point () is an interior point, while the value of a Hermite interpolation function with the form of weighted combination will be used if it is an exterior point. Hermite interpolation needs the first-order derivatives of the interpolated function which is not directly given in scatted data, so this paper also gives the approximate derivative at every scatted point using local radial basis function interpolation. And numerical tests indicate that the proposed piecewise bivariate Hermite interpolations are economic and have good approximation capacity.

Journal ArticleDOI
TL;DR: In this paper, the authors study several interpolation schemes that are adequate for the yield curve construction, and pay attention to their stability under sequential and parallel perturbations, and find that Hagan and West monotone convex interpolation, tension splines and some Monotone Hermite spline interpolation do not always create stable yield curves.
Abstract: We study several interpolation schemes that are adequate for the yield curve construction, and pay attention to their stability under sequential and parallel perturbations. It is found that Hagan and West monotone convex interpolation, tension splines and some monotone Hermite spline don’t always create stable yield curves. A specific monotone Hermite spline interpolation is however stable.

Journal ArticleDOI
TL;DR: This paper focuses on Tschirnhausen cubic as the only one Pythagorean hodograph cubic and it is proved that the approximation order is 3.1, and analyzes the shape of TC-biarcs, which is an asymptotical behaviour of the conversion of an arbitrary planar curve with well defined tangent vectors everywhere to a C^1 PH cubic spline curve.

Journal ArticleDOI
TL;DR: Two algorithms for interpolating by weighted cubic splines with the automatic choice of the shape-controlling parameters (weights) are developed, one of them preserves the monotonicity of the data, while the other preserves the convexity.
Abstract: Algorithms for interpolating by weighted cubic splines are constructed with the aim of preserving the monotonicity and convexity of the original discrete data. The analysis performed in this paper makes it possible to develop two algorithms with the automatic choice of the shape-controlling parameters (weights). One of them preserves the monotonicity of the data, while the other preserves the convexity. Certain numerical results are presented.

Journal ArticleDOI
TL;DR: In this article, a linear and non-linear diffusion-dispersion models involving fluid flow through porous cylindrical particles are solved using orthogonal collocation on finite elements with cubic Hermite as basis.

Journal ArticleDOI
TL;DR: In this article, the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval is formulated and analyzed.
Abstract: We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

Patent
Jeffrey A. Bolz1
16 Sep 2013
TL;DR: In this paper, a cubic preprocessor is coupled to a central processing unit that formats the cubic Bezier curve to provide a formatted cubic curve having quadrilateral control points corresponding to a mathematically simple cubic curve.
Abstract: A graphics processing system includes a central processing unit that processes a cubic Bezier curve corresponding to a filled cubic Bezier path. Additionally, the graphics processing system includes a cubic preprocessor coupled to the central processing unit that formats the cubic Bezier curve to provide a formatted cubic Bezier curve having quadrilateral control points corresponding to a mathematically simple cubic curve. The graphics processing system further includes a graphics processing unit coupled to the cubic preprocessor that employs the formatted cubic Bezier curve in rendering the filled cubic Bezier path. A rendering unit and a display cubic Bezier path filling method are also provided.

01 Jan 2013
TL;DR: An overview of the theory of interpolation and its applications in numerical analysis can be found in this article, where the authors focus on cubic splines interpolation with simulations in Matlab™.
Abstract: The paper is an overview of the theory of interpolation and its applications in numerical analysis. It specially focuses on cubic splines interpolation with simulations in Matlab™.

Journal ArticleDOI
TL;DR: The monotone preserving r(t)t method as mentioned in this paper applies shape-preserving cubic Hermite interpolation to the log capitalisation function to ensure positive and continuous forward curves.
Abstract: This paper presents a method for interpolating yield curve data in a manner that ensures positive and continuous forward curves. As shown by Hagan and West (2006), traditional interpolation methods suffer from problems: they posit unreasonable expectations, or are not necessarily arbitrage-free. The method presented in this paper, which we refer to as the “monotone preserving r(t)t method", stems from the work done in the field of shape preserving cubic Hermite interpolation, by authors such as Akima (1970), de Boor and Swartz (1977), and Fritsch and Carlson (1980). In particular, the monotone preserving r(t)t method applies shape preserving cubic Hermite interpolation to the log capitalisation function. We present some examples of South African swap and bond curves obtained under the monotone preserving r(t)t method.

Journal ArticleDOI
TL;DR: Stability analysis of the numerical method based on parametric cubic splines for solving the cubic nonlinear Schrodinger equation has been carried out and the method is shown to be unconditionally stable.

Journal ArticleDOI
TL;DR: A construction of a cubic Bézier spline surface that interpolates prescribed spatial points and the corresponding normal directions of tangent planes is proposed and the interpolant minimizes Willmore energy functional.
Abstract: In this paper, a construction of a cubic Bezier spline surface that interpolates prescribed spatial points and the corresponding normal directions of tangent planes is proposed. Boundary curves of each triangular patch minimize the approximated strain energy. A comparison of optimal boundary curves is given. The interpolant minimizes Willmore energy functional. Some numerical examples and applications of the interpolation scheme are presented: surface approximation, hole filling and condensation of parameters.


Journal ArticleDOI
TL;DR: This work presents a multi-level quasi-interpolation method which directly uses normal vectors to construct non-zero constraints and avoids solving any linear system, a common step of variational surface reconstruction, and leads to a fast and stable surface reconstruction from scattered points.
Abstract: Based on the Hermite variational implicit surface reconstruction presented in Pan et al. (Science in China Series F: Information Sciences 52(2):308---315, 2009), we propose a multi-level interpolation method to overcome the problems resulted from using compactly supported radial basis functions (CSRBFs). In addition, we present a multi-level quasi-interpolation method which directly uses normal vectors to construct non-zero constraints and avoids solving any linear system, a common step of variational surface reconstruction, and leads to a fast and stable surface reconstruction from scattered points. With adaptive support size, our approach is robust and can successfully reconstruct surfaces on non-uniform and noisy point sets. Moreover, as the computation of quasi-interpolation is independent for each point, it can be easily parallelized on multi-core CPUs.

Posted Content
TL;DR: In this article, the Hermite problem has been approached finding a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality, and a multidimensional continued fraction is derived from a modification of the Jacobi algorithm, which is proved periodic if and only if the inputs are cubic irrationals.
Abstract: In this paper, the Hermite problem has been approached finding a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In other words, the problem of writing cubic irrationals as a periodic sequence of rational or integer numbers has been solved. In particular, a periodic multidimensional continued fraction (with pre--period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences. This multidimensional continued fraction is derived from a modification of the Jacobi algorithm, which is proved periodic if and only if the inputs are cubic irrationals. Moreover, this representation provides simultaneous rational approximations for cubic irrationals.

Journal ArticleDOI
TL;DR: The asymptotic convergence of cubic Hermite collocation method in continuous time for the parabolic partial differential equation is established of order Oh^2.

Journal ArticleDOI
TL;DR: In this article, the spline split quaternion interpolation on hyperbolic sphere in Minkowski space using split Quaternions and metric Lorentz is presented.
Abstract: Spherical spline quaternion interpolation has been done on sphere in Euclidean space using quaternions. In this paper, we have done the spline split quaternion interpolation on hyperbolic sphere in Minkowski space using split quaternions and metric Lorentz. This interpolation curve is called spherical spline split quaternion interpolation in Minkowski space (MSquad).