Showing papers on "Monotone cubic interpolation published in 2012"

Shape-preserving curve interpolation

Abstract: This work is a contribution towards the graphical display of 2D data when they are convex, monotone and positive. A piecewise rational function in a cubic/cubic form is proposed, which, in each interval, involves four free parameters in its construction. These four free parameters have a direct geometric interpretation, making their use straightforward. Illustrations of their effect on the shape of the rational function are given. Two of these free parameters are constrained to preserve the shape of convex, monotone and positive data, while the other two parameters are utilized for the modification of positive, monotone and convex curves to obtain a visually pleasing curve. The problem of shape preservation of data lying above a line is also discussed. The method that is presented applies equally well to data or data with derivatives. The developed scheme is computationally economical and pleasing. The error of rational interpolating function is also derived when the arbitrary function being interpolated ...

A rational Krylov method based on Hermite interpolation for the nonlinear eigenvalue problem

Abstract: This paper proposes a new rational Krylov method for solving the nonlinear eigenvalue problem: $A(\lambda)x = 0$. The method approximates $A(\lambda)$ by Hermite interpolation where the degree of the interpolating polynomial and the interpolation points are not fixed in advance. It uses a companion-type reformulation to obtain a linear generalized eigenvalue problem (GEP). To this GEP we apply a rational Krylov method that preserves the structure. The companion form grows in each iteration and the interpolation points are dynamically chosen. Each iteration requires a linear system solve with $A(\sigma)$, where $\sigma$ is the last interpolation point. The method is illustrated by small- and large-scale numerical examples. In particular, we illustrate that the method is fully dynamic and can be used as a global search method as well as a local refinement method. In the last case, we compare the method to Newton's method and illustrate that we can achieve an even faster convergence rate.

Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations

Abstract: Numerical solutions of one-dimensional heat and advection-diffusion equations are obtained by collocation method based on cubic B-spline. Usual finite difference scheme is used for time and space integrations. Cubic B-spline is applied as interpolation function. The stability analysis of the scheme is examined by the Von Neumann approach. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.

Construction of interpolation splines minimizing semi-norm in \(W_{2}^{(m,m-1)}(0,1)\) space

Abstract: In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in a Hilbert space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for polynomials of degree m−2 and e −x . Also some numerical results are presented.

Cubic hermite and cubic spline fractal interpolation functions

Abstract: Despite that the spline theory is a well studied topic, its relationship with the fractal theory is novel. Fractal approach offers a single specification for a large class of interpolants of which the classical spline is a particular member, and hence possesses considerable flexibility in the choice of an interpolant. The explicit construction of a C1-cubic Hermite fractal interpolation function (FIF) is introduced in the present work. If slopes at knot points are not known, then they are calculated through solution of a suitable linear system of equations so as to have C2 global smoothness for the resulting cubic FIF. Thus, the present method generalizes the classical C1-cubic Hermite and C2-cubic spline interpolants simultaneously, and offers a new approach to the development of cubic spline FIF in contrast to the construction through moments by Chand and Kapoor [SIAM J. Numer. Anal., 44(2), (2006), pp. 655-676]. It is shown that, for appropriate values of vertical scaling factors involved in the definition, developed C1-cubic Hermite FIF converges uniformly to the data generating function Φ ∈ C4 at least as rapidly as fourth power of the mesh norm approaches zero.

An approach to geometric interpolation by Pythagorean-hodograph curves

Abstract: The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree n is studied independently of the dimension d???2. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.

Natural cubic spline coalescence hidden variable fractal interpolation surfaces

Abstract: Fractal interpolation functions provide a new light to the natural deterministic approximation and modeling of complex phenomena. The present paper proposes construction of natural cubic spline coalescence hidden variable fractal interpolation surfaces (CHFISs) over a rectangular grid through the tensor product of univariate bases of cardinal natural cubic spline coalescence hidden variable fractal interpolation functions (CHFIFs). Natural cubic CHFISs are self-affine or non-self-affine in nature depending on the IFS parameters of univariate natural cubic spline CHFIFs. An upper bound of the error between the natural cubic spline blended coalescence fractal interpolant and the original function is deduced. Convergence of the natural cubic CHFIS to the original function , and their derivatives are deduced. The effects free variables, constrained free variables and hidden variables are discussed on the natural cubic spline CHFIS with suitably chosen examples.

From Taylor interpolation to Hermite interpolation via duality

Abstract: The present work concerns W-spaces, that is, spaces which permit Taylor interpolation on a given interval. We introduce the critical length of any given W-space E as the supremum of all positive h ensuring that E permits Hermite interpolation ( i.e. , E is an Extended Chebyshev space) on any subinterval of length h . The critical length may be equal to 0, but it is always positive if the interval is closed and bounded. Any W-space is allocated to a dual space. When the dual space is a W-space in turn, we can take advantage of its presence to calculate the critical length. The notion of critical length was first introduced in [3] for null spaces of linear differential operator with constant coefficients. As a special case, the use of duality gives new insights into the practical expressions to obtain the critical length of such null spaces.

Numerical Simulation of Plane Crack Using Hermite Cubic Spline Wavelet

Abstract: Two-dimensional wavelet-based numerical approximation using Hermite cubic spline wavelet on the interval (HCSWI) is proposed to solve stress intensity factors (SIFs) of plate structures. The good localization property of wavelets is used to approximate displacement fields by multi-scale bases of HCSWI. Example computations are performed for plates with a central crack and double edge cracks. The numerical results prove that, compared with the conventional finite element method and the analytical solutions, the new procedure are efficient in both its accuracy and its reduction of degree of freedoms (DOFs).

Phase lag analysis of variational integrators using interpolation techniques

Abstract: In the present work we derive higher order variational integrators and combine them with phase lag properties for the numerical integration of systems with oscillatory solutions. The discrete Lagrangian in any time interval is defined as a weighted sum of the evaluation of the continuous Lagrangian at intermediate time nodes. The expressions used for configurations and velocities use linear interpolation, cubic spline interpolation or interpolation via trigonometric functions. The new methods depend on a frequency, which needs to be chosen appropriately. Results show that the energy error of the integration method is decreased for good frequency estimates. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Adaptive image interpolation by cardinal splines in piecewise constant tension

Abstract: The cardinal spline in tension is modified to allow for different tensions in different sampling intervals. Varying the tension in proportion to an index of sharp change in image brightness, we obtain image interpolation results with less ringing artifacts compared to those by the cubic spline interpolation.

C-shaped G2 Hermite interpolation with circular precision based on cubic PH curve interpolation

Abstract: Based on the technique of C-shaped G^1 Hermite interpolation by a cubic Pythagorean-hodograph (PH) curve, we present a simple method for C-shaped G^2 Hermite interpolation by a rational cubic Bezier curve. The method reproduces a circular arc when the input data come from it. Both the Bezier control points, which have a well-understood geometrical meaning, and the weights of the resulting rational cubic Bezier curve are expressed in explicit form. We test our method with many numerical examples, and some of them are presented here to demonstrate the properties of our method. The comparison between our method and a previous method is also included.

Hermite Geometric Interpolation by Rational Bézier Spatial Curves

Abstract: Polynomial geometric interpolation by parametric curves has become one of the standard techniques for interpolation of geometric data. An obvious generalization leads to rational geometric interpolation schemes, which are a much less investigated research topic. The aim of this paper is to present a general framework for Hermite geometric interpolation by rational Bezier spatial curves. In particular, cubic $G^2$ and quartic $G^3$ interpolations are analyzed in detail. Systems of nonlinear equations are derived in a simplified form, and the existence of admissible solutions is studied. For the cubic case, geometric conditions implying solvability of the nonlinear system are also stated. The asymptotic analysis is done in both cases, and optimal approximation orders are proved. Numerical examples are given, which confirm the theoretical results.

Efficient cubic spline interpolation implemented with FIR filters

Abstract: Classical Cubic spline interpolation needs to solve a set of equations of high dimension. In this work we show how to compute the interpolant using a FIR digital filter, with a reduced number of operations per interpolated point and high accuracy. Additionally, the computation can be made on real time as the signal samples are acquired. Following this approach, we show how to obtain easily the derivatives of the interpolant in a similar way, and also signal approximations to reduce the oscillations that appear when using high order splines. These techniques are very well suited to compute continuous representations of image contours on closed shapes and to find its curvature and singularities.

Quarter-sweep modified SOR iterative algorithm and cubic spline basis for the solution of second order two-point boundary value problems

Abstract: The aim of this study is to describe the formulation of Quarter-Sweep Modified Successive Over-Relaxation (QSMSOR) iterative method using cubic polynomial spline scheme for solving second order two-point linear boundary value problems. To solve the problems, a linear system will be constructed via discretization process by using cubic spline approximation equation. Then the generated linear system has been solved using the proposed QSMSOR iterative method to show the superiority over Full-Sweep Modified Successive Over-Relaxation (FSMSOR) and Half-Sweep Modified Successive Over-Relaxation (HSMSOR) methods. Computational results are provided to illustrate the effectiveness of the proposed method.

Improving empirical mode decomposition with an optimized piecewise cubic Hermite interpolation method

Abstract: Empirical mode decomposition (EMD) is an adaptive method for analyzing non-stationary time series derived from linear and nonlinear systems. But the upper and lower envelopes fitted by cubic spline (CS) interpolation may often occur overshoots. In this paper, a novel envelope fitting method based on the optimized piecewise cubic Hermite (OPCH) interpolation is developed. Taking the difference between extreme as the cost function, chaos particle swarm optimization (CPSO) method is used to optimize the derivatives of the interpolation nodes. The flattest envelope with the optimized derivatives can overcome the overshoots well. Some numerical experiments conclude this paper, and comparisons are carried out with the classical EMD.

Obtaining a banded system of equations in complete spline interpolation problem via B-spline basis

Abstract: We study the problem of interpolation by a complete spline of 2n − 1 degree given in B-spline representation. Explicit formulas for the first nand the last ncoefficients of B-spline decomposition are found. It is shown that other B-spline coefficients can be computed as a solution of a banded system of an equitype linear equations.

Numerical differentiation using spline functions

Abstract: We consider the application of splines with minimum-norm derivative in numerical differentiation. Tabular functions are approximated by a cubic spline with a piecewise-continuous second derivative, which ensures high-accuracy evaluation of the derivative.

The Numerical Solution of Heat Problem Using Cubic B-Splines

Abstract: This paper discusses solving one of the important equations in Physics; which is the one-dimensional heat equation. For that purpose, we use cubic B-spline fin ite elements within a Collocation method. The scheme of the method is presented and the stability analysis is investigated by considering Fourier stability method. On the other hand, a comparative study between the numerical and the analytic solution is illustrated by the figure and the tables. The results demonstrate the reliability and the efficiency of the method.

Hermite interpolation using ph curves with undetermined junction points

Abstract: Representing planar Pythagorean hodograph (PH) curves by the complex roots of their hodographs, we standardize Farouki's double cubic method to become the undetermined junction point (UJP) method, and then prove the generic existence of solutions for general C 1 Hermite interpolation problems. We also extend the UJP method to solve C 2 Hermite interpolation problems with multiple PH cubics, and also prove the generic existence of solutions which consist of triple PH cubics with C 1 junction points. Further generalizing the UJP method, we go on to solve C 2 Hermite interpolation problems using two PH quintics with a C 1 junction point, and we also show the possibility of applying the modied UJP method to G 2 ( C 1 ) Hermite interpolation.