Showing papers on "Monotone cubic interpolation published in 2014"
TL;DR: This paper addresses two interrelated problems concerning the planar three degree-of-freedom motion of a vehicle, namely, the path planning problem and the guidance problem, and derives a new κ-exponentially stable integral LOS guidance law, capable of eliminating the effect of constant external disturbances for straight-line path following.
Abstract: This paper addresses two interrelated problems concerning the planar three degree-of-freedom motion of a vehicle, namely, the path planning problem and the guidance problem. The monotone cubic Hermite spline interpolation (CHSI) technique by Fritsch and Carlson is employed to design paths that provide the user with better shape control and avoid wiggles and zigzags between the two successive waypoints. The conventional line-of-sight (LOS) guidance law is modified by proposing a time-varying equation for the lookahead distance, which is a function of the cross-track error. This results in a more flexible maneuvering behavior that can contribute to reaching the desired path faster as well as obtaining a diminished oscillatory behavior around the desired path. The guidance system along with a heading controller form a cascaded structure, which is shown to be $\kappa$ -exponentially stable when the control task is to converge to the path produced by the aforementioned CHSI method. In addition, the issue of compensating for the sideslip angle $\beta$ is discussed and a new $\kappa$ -exponentially stable integral LOS guidance law, capable of eliminating the effect of constant external disturbances for straight-line path following, is derived.
267 citations
TL;DR: A collocation finite difference scheme based on new cubic trigonometric B-spline is developed and analyzed for the numerical solution of a one-dimensional hyperbolic equation (wave equation) with non-local conservation condition and is shown to be unconditionally stable using the von Neumann (Fourier) method.
72 citations
TL;DR: In this article, the authors developed a new class of rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form of cubic polynomials involving two shape parameters.
Abstract: Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $$\mathcal C ^1$$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $$\frac{p_i(x)}{q_i(x)},$$ where $$p_i(x)$$ and $$q_i(x)$$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $$\mathcal C ^2$$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.
67 citations
01 Nov 2014
TL;DR: The simulation results show that the cubic B-Spline curve recursive formula of Taylor development of first-order derivation of two order in the interpolation cycle meet the high-speed and high-accuracy NC machine tool require-ments of CNC systems.
Abstract: Based on cubic B-Spline curve mathematical properties, theoretical analysis the cubic B-Spline curve recursive formula of Taylor development of first-order, derivation of two order in the interpolation cycle under the condition of certain interpolation increment only and interpolation speed, change the interpolation increments can be amended cubic times B-Spline curves purpose The simulation results show that meet the high-speed and high-accuracy NC machine tool require-ments of CNC systems.
43 citations
TL;DR: Two interpolation-based monotone schemes for the anisotropic diffusion problems on unstructured polygonal meshes are proposed through the linearity-preserving approach, and have the second-order accuracy for the solution and first-order for its gradient on severely distorted meshes.
38 citations
TL;DR: Method to determine optimal value of parameters which generate minimum error in approximation is presented and applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations, and to Empirical Mode Decomposition to illustrate practical usefulness of the results.
31 citations
TL;DR: In this paper, three numerical techniques based on cubic Hermite spline functions are presented for the solution of the Lane-Emden equation, and some properties of Hermite spines are used to reduce the problem of solving the Lemden equation to algebraic equations.
Abstract: Three numerical techniques based on cubic Hermite spline functions are presented for the solution of Lane–Emden equation. Some properties of Hermite splines are presented and are utilized to reduce the solution of Lane–Emden equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of these techniques. Copyright © 2013 John Wiley & Sons, Ltd.
30 citations
TL;DR: In this article, a cubic spline radial basis function (RBF) interpolation scheme was used to compute European and American option prices under the jump-diffusion model using the RBF interpolation algorithm.
Abstract: This paper will demonstrate how European and American option prices can be computed under the jump-diffusion model using the radial basis function (RBF) interpolation scheme. The RBF interpolation scheme is demonstrated by solving an option pricing formula, a one-dimensional partial integro-differential equation (PIDE). We select the cubic spline radial basis function and adopt a simple numerical algorithm (Briani et al. in Calcolo 44:33–57, 2007) to establish a finite computational range for the improper integral of the PIDE. This algorithm reduces the truncation error of approximating the improper integral. As a result, we are able to achieve a higher approximation accuracy of the integral with the application of any quadrature. Moreover, we a numerical technique termed cubic spline factorisation (Bos and Salkauskas in J Approx Theory 51:81–88, 1987) to solve the inversion of an ill-conditioned RBF interpolant, which is a well-known research problem in the RBF field. Finally, our numerical experiments show that in the European case, our RBF-interpolation solution is second-order accurate for spatial variables, while in the American case, it is second-order accurate for spatial variables and first-order accurate for time variables.
28 citations
TL;DR: A rational bi-cubic function involving six shape parameters is presented for shape preserving interpolation problem for visualization of 3D positive data which is an extension of piecewise rational function in the form of cubic/quadratic involving three shape parameters.
24 citations
TL;DR: The scheme is based on the usual finite difference scheme to discretize the time derivative while cubic B-spline is used as an interpolation function in the space dimension and is shown to be unconditionally stable using the von Neumann method.
Abstract: In this paper, a numerical method for the solution of a strongly coupled reaction-diffusion system, with suitable initial and Neumann boundary conditions, by using cubic B-spline collocation scheme on a uniform grid is presented. The scheme is based on the usual finite difference scheme to discretize the time derivative while cubic B-spline is used as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann method. The accuracy of the proposed scheme is demonstrated by applying it on a test problem. The performance of this scheme is shown by computing and error norms for different time levels. The numerical results are found to be in good agreement with known exact solutions.
23 citations
TL;DR: The C^1 Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described.
TL;DR: A new type of C^1-rational cubic spline Fractal Interpolation Function (FIF) for convexity preserving univariate interpolation, well suited for the approximation of a convex function @F whose derivative is continuous but has varying irregularity.
TL;DR: An optimized version of the Akima's interpolation method for experimental data fitting, at the end-points, of the Catmull–Rom's cubic spline is proposed, and the error estimate is improved.
TL;DR: The proposed monotonicity preserving rational FIF subsumes and supplements a classical monotonic rational cubic spline and has O ( h 4 ) accuracy, provided the original function belongs to C 4 ( I ) and the parameters involved in the FIF are appropriately chosen.
TL;DR: In this paper a cubic Hermite interpolation for fuzzy data is presented and then it is generalized to piecewise cubic Hermites interpolation, which is constructed under some weaker conditions than of HermITE interpolation.
Abstract: In this paper a cubic Hermite interpolation for fuzzy data is presented and then it is generalized to piecewise cubic Hermite interpolation. Moreover an error bound is given for piecewise cubic Hermite interpolation. The piecewise cubic Hermite interpolation is constructed under some weaker conditions than of Hermite interpolation. The results of this paper are illustrated by some examples.
TL;DR: In this paper, the authors proposed a new C 1 rational cubic spline interpolant of the form cubic/quadratic with three shape parameters to preserve the geometric properties of the given data sets.
Abstract: This study proposes new C 1 rational cubic spline interpolant of the form cubic/quadratic with three shape parameters to preserves the geometric properties of the given data sets. Sufficient conditions for the positivity and data constrained modeling of the rational interpolant are derived on one parameter while the remaining two parameters can further be utilized to change and modify the final shape of the curves. The sufficient conditions ensure the existence of positive and constrained rational interpolant. Several numerical results will be presented to test the capability of the proposed rational interpolant scheme. Comparisons with the existing scheme also have been done. From all numerical results, the new rational cubic spline interpolant gives satisfactory results.
TL;DR: In this paper, the Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated.
Abstract: In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x0, xN], the interpolating Cubic Spline SFIF and their derivatives converge respectively to the data generating function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2-j+ϵ(0 < ϵ < 1), j = 0, 1, 2, as the norm h of the partition of [x0, xN] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives.
TL;DR: In this article, the rational cubic spline with three parameters has been used to preserve the convexity of the data and the sufficient condition for rational interpolant to be convex on entire subinterval has been developed.
Abstract: This study is a continuation of our previous paper. The rational cubic spline with three parameters has been used to preserves the convexity of the data. The sufficient condition for rational interpolant to be convex on entire subinterval will be developed. The constraint will be on one of the parameter with data dependent meanwhile the other are free parameters and will determine the final shape of the convex curves. Several numerical results will be presented to test the capability of the proposed rational interpolant scheme. Comparisons with the existing scheme also have been done. From all numerical results, the new rational cubic spline interpolant gives satisfactory results.
19 Dec 2014
TL;DR: By PCS, the Runge function is effectively approximated without causing oscillation, and numerical experiments substantiate the efficacy and accuracy of PCS.
Abstract: The Runge phenomenon illustrates that equidistant polynomial interpolation of the Runge function will cause wild oscillation near the endpoints of the interpolation interval as the order of the interpolation polynomial increases. In this paper, the pseudo inverse cubic spline (PCS) is presented to accurately approximate the Runge function at equidistant interpolation nodes and solve the problem of Runge phenomenon. PCS is constructed by employing the right pseudo inverse to figure out the minimum norm solution of the cubic spline's second-order derivatives. Thus, unlike the traditional cubic splines that additionally rely on endpoint constraints, PCS only employs the information of interpolation nodes. By PCS, the Runge function is effectively approximated without causing oscillation. Numerical experiments substantiate the efficacy and accuracy of PCS.
25 Mar 2014
TL;DR: In this paper, the notions of cubic subalgebras and cubic lters in CI-algeses are introduced, and related properties are investigated, and conditions for a cubic set to be a cubic lter are provided.
Abstract: . The notions of cubic subalgebras and cubic lters inCI-algebras are introduced, and related properties are investigated.Characterizations of cubic subalgebras are considered. Conditionsfor a cubic set to be a cubic lter are provided. 1. IntroductionAs a generalization of a BCK-algebra, Kim and Kim [6] introducedthe notion of a BE-algebra, and investigated several properties. Thenotion of CI-algebras is introduced by Meng [8] as a generalization ofBE-algebras. Filter theory and properties in CI-algebras are studiedby Kim [7], Meng [9] and Piekart et al. [10]. Fuzzy sets, which wereintroduced by Zadeh [11], deal with possibilistic uncertainty, connectedwith imprecision of states, perceptions and preferences. Based on the(interval-valued) fuzzy sets, Jun et al. [3] introduced the notion of (inter-nal, external) cubic sets, and investigated several properties. Jun et al.applied the notion of cubic sets to BCK=BCI-algebras (see [1, 2, 4, 5]).In this paper, we discuss the notions of cubic subalgebras and cubic lters in CI-algebras. We investigated several related properties. Weconsider characterizations of cubic subalgebras. We provide conditionsfor a cubic set to be a cubic lter.
TL;DR: In this paper, the authors developed a numerical method based on cubic polynomial spline approximations to solve a generalized Black-Scholes equation and showed that the matrix associated with the discrete operator is an M-matrix, which ensures that the scheme is maximum norm stable.
Abstract: We develop a numerical method based on cubic polynomial spline approximations to solve a a generalized Black-Scholes equation We apply the implicit Euler method for the time discretization and a cubic polynomial spline method for the spatial discretization We show that the matrix associated with the discrete operator is an M-matrix, which ensures that the scheme is maximum-norm stable It is proved that the scheme is second-order convergent with respect to the spatial variable Numerical examples demonstrate the stability, convergence, and robustness of the scheme
TL;DR: This paper uses discrete cubic spline based on central differences to obtain approximate solution of a second-order boundary value problem and shows that the method is of order 4 if a parameter takes a specific value, else it is of orders of order 2.
Abstract: In this paper, we use discrete cubic spline based on central differences to obtain approximate solution of a second-order boundary value problem. It is shown that the method is of order 4 if a parameter takes a specific value, else it is of order 2. Two numerical examples are included to illustrate our method as well as to compare the performance with other numerical methods proposed in the literature.
TL;DR: In this article, two types of cubic spline are used for data interpolation and fitting the data to produce the interpolating curves or fitting curves are important in O&G industry.
Abstract: Interpolation and fitting the data to produce the interpolating curves or fitting curves are important in Oil and Gas (O&G) industry. Data interpolation is useful for scientific visualization for data interpretation. One of the efficient methods for data interpolation is cubic spline function. In this paper two types of cubic spline will be used for data interpolation. The first one is cubic spline interpolation with
01 Dec 2014
TL;DR: A novel dynamic time warping based on cubic spline interpolation (SIDTW) to improve the performance and demonstrates that in most cases this approach not only can produce much less singularities and obtain the best warping path with shorter length but also is an alternative version of DTW when time series datasets are not suitable for DTW to be measured.
Abstract: Dynamic time warping (DTW) and derivative dynamic time warping (DDTW) are two robust distance measures for time series, which allows similar shapes to match even if they are out of phase in the time axis In this paper, we propose a novel dynamic time warping based on cubic spline interpolation (SIDTW) to improve the performance The derivative of every point of time series is calculated by cubic spline interpolation and is used to replace the estimated derivatives in DDTW After interpolation we use derivative-based sequences to represent the original time series, which is better to describe the trend of the original time series and more reasonable to warp Meanwhile, we empirically point out that the quality of similarity measure for the three warping methods is nothing to do with the amount of warping We experimentally perform the proposed method and compare with the existing ones, which demonstrates that in most cases our approach not only can produce much less singularities and obtain the best warping path with shorter length but also is an alternative version of DTW when time series datasets are not suitable for DTW to be measured
TL;DR: In this article, a family of quasi-interpolants based on bivariate cubic super splines defined on triangulations with a Powell-Sabin refinement was proposed, whose spline coefficients only depend on a set of local function values.
Abstract: By using the polarization identity, we propose a family of quasi-interpolants based on bivariate $${\fancyscript{C}}^1$$
cubic super splines defined on triangulations with a Powell–Sabin refinement. Their spline coefficients only depend on a set of local function values. The quasi-interpolants reproduce cubic polynomials and have an optimal approximation order.
10 Jul 2014
TL;DR: In this article, a smooth curve interpolation scheme for positive data is developed, which uses rational cubic Ball representation and conditions are derived for preserving positivity and C1 continuity for a number of numerical experiments.
Abstract: A smooth curve interpolation scheme for positive data is developed. Conditions have been incorporated into this scheme to preserve the shape of the data lying above a line. This scheme uses rational cubic Ball representation. Conditions are derived for preserving positivity and C1 continuity. The outputs from a number of numerical experiments are presented.
TL;DR: In this article, the authors investigate the advantages in terms of shape preservation and computational efficiency of calculating univariate cubic spline fits using a steepest-descent algorithm to minimize a global data-fitting functional under a constraint implemented by a local analysis-based interpolating-spline algorithm on 5-node windows.
Abstract: $$L^1$$ splines have been under development for interpolation and approximation of irregular geometric data We investigate the advantages in terms of shape preservation and computational efficiency of calculating univariate cubic $$L^1$$ spline fits using a steepest-descent algorithm to minimize a global data-fitting functional under a constraint implemented by a local analysis-based interpolating-spline algorithm on 5-node windows Comparison of these locally calculated $$L^1$$ spline fits with globally calculated $$L^1$$ spline fits previously reported in the literature indicates that the locally calculated spline fits preserve shape on the average slightly better than the globally calculated spline fits and are computationally more efficient because the locally-calculated-spline-fit algorithm can be parallelized
TL;DR: It was indicated that the local control of interpolating function by using rational cubic spline with three parameters works well and it is comparable with the established existing scheme.
Abstract: This paper discussed the local control of interpolating function by using rational cubic spline (cubic/quadratic) with three parameters originally proposed by the authors. The rational spline has continuity. The bounded properties of the rational cubic interpolants and shape controls of the rational interpolants are discussed in detail. The value control, inflection point control, and convexity control at a point by using the proposed rational cubic spline are constructed. Several numerical results are presented to show the capability of the method. Numerical comparisons with the existing scheme are also further elaborated. From the results, it was indicated that the scheme works well and it is comparable with the established existing scheme.
TL;DR: A novel correction technique employing a cubic spline interpolation is proposed for inter-channel sample -time error compensation and the experimental results showed that the correction technique is effective to attenuate the spurious spurs and improve the dynamic performance of the system.
Abstract: Sample -time errors can greatly degrade the dynamic range of a time-interleaved sampling system . In this paper, a novel correction technique employing a cubic spline interpolation is proposed for inter-channel sample -time error compensation. The cubic spline interpolation compensation filter is developed in the form of a finiteimpulse response (FIR) filter structure. The correction method of the interpolation compensation filter coefficients is deduced. A 4GS/s two-channel, time-interleaved ADC prototype system has been implemented to evaluate the performance of the technique. The experimental results showed that the correction technique is effective to attenuate the spurious spurs and improve the dynamic performance of the system.
TL;DR: An algorithm using the cubic spline interpolation on the finite difference method to solve the Bratu-type equation is proposed and implemented.
Abstract: In this paper, we propose an algorithm using the cubic spline interpolation on the finite difference method to solve the Bratu-type equation. The algorithm has been successfully implemented. Numerical results are also given to demonstrate the validity and the applicability of the proposed algorithm. The results we obtained show that the proposed algorithm perform better than some existing methods in the literature.