Linear Denominator Research Articles

Positivity-Preserving Rational Cubic Fractal Interpolation Function Together with Its Zipper Form

In this paper, a novel class of rational cubic fractal interpolation function (RCFIF) has been proposed, which is characterized by one shape parameter and a linear denominator. In interpolation for shape preservation, the proposed rational cubic fractal interpolation function provides a simple but effective approach. The nature of shape preservation of the proposed rational cubic fractal interpolation function makes them valuable in the field of data visualization, as it is crucial to maintain the original data shape in data visualization. Furthermore, we discussed the upper bound of error and explored the mathematical framework to ensure the convergence of RCFIF. Shape parameters and scaling factors are constraints to obtain the desired shape-preserving properties. We further generalized the proposed RCFIF by introducing the concept of signature, giving its construction in the form of a zipper-rational cubic fractal interpolation function (ZRCFIF). The positivity conditions for the rational cubic fractal interpolation function and zipper-rational cubic fractal interpolation function are found, which required a detailed analysis of the conditions where constraints on shape parameters and scaling factor lead to the desired shape-preserving properties. In the field of shape preservation, the proposed rational cubic fractal interpolation function and zipper fractal interpolation function both represent significant advancement by offering a strong tool for data visualization.

Local competing in interpolation problems

A simple example illustrates the insufficiency of the known approaches to interpolation in the problem of recovering a function from a few given specific values that clearly convey the form. A local choice between polynomial and rational local interpolants, which minimizes the local interpolant’s errors at the nearest external nodes from one or different sides, complements the known approaches. It combines the extreme computational simplicity of local interpolants with the thorought selection of them. The principles of constructing the algorithm are formulated in general terms for mappings of metric spaces. They provide accurate (with rare exceptions) reconstruction of mappings that locally coincide with some of the given possible interpolants. In the one-dimensional case, the two-stage algorithm guarantees the continuity of the interpolant and accurately reconstructs polynomials of small degree, simple rational functions with a linear denominator, and broken lines of long links with knots at the ends when these requirements do not contradict each other. An additional parameter allows you to replace the exact restoration of polylines with the required smoothness of interpolation.

Rational Integrals of a Natural Mechanical System on the 2-Torus

Under study are the problems of existence of the first integrals rational in momenta for a natural mechanical system on the 2-torus. We prove that a rational integral with a linear numerator and denominator reduces to a linear integral. Considering the case of a quadratic numerator and linear denominator, we obtain an analogous result under some additional assumptions on the coefficients of the first integral.

Finit element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for linear convex quadrilaterals of cubic order Serendipity and Lagrange families

This paper presents an explicit integration scheme to compute the stiffness matrix of twelve node and sixteen node linear convex quadrilateral finite elements of Serendipity and Lagrange families using an explicit integration scheme and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique, In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties matrices and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+) in the bivariates over a 2-square (-1 ) with the nodes on the boundary and in the interior of this simple domain. The finite elements up to cubic order have nodes only on the boundary for Serendipity family and the finite elements with boundary as well as some interior nodes belong to the Lagrange family. The first order element is the bilinear convex quadrilateral finite element which is an exception and it belongs to both the families. We have for the present ,the cubic order finite elements which havee 12 boundary nodes at the nodal coordinates {(-1,-1),(1,-1),(1,1),(-1,1),(-1/3,-1), (1/3,-1),(1,-1/3),(1,1/3),(1/3,1),(-1/3,1),(-1,1/3),(-1,-1/3)} and the four interoior nodal coordinates at the points (-1/3,-1/3),(1/3,-1/3),(1/3,1/3),(-1/3,1/3)} in the local parametric space ( In this paper, we have computed the integrals of local derivative products with linear denominator (4+) in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be then applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all 12-node and 16-node linear(straight edge) convex quadrilaterals for a polygonal domain which provides the nodal coordinates and the element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is an unknown physical variable and in with Dirichlet boundary conditions over the convex polygonal domain.

Finite element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for nine node linear convex quadrilateral of Lagrange family

This paper presents an explicit integration scheme to compute the stiffness matrix of a nine node linear convex quadrilateral element of Lagrange family using symbolic mathematics and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique , In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+ ) in the bivariates over a nine node 2-square (-1 ) with nodes at the points {(-1,-1),(1,-1),(1,1),(-1,1),(0,-1),(1,0).(0,1),(- 1,0),(0,0)} in local parametric space. In this paper, we have computed these integrals in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all nine node linear convex quadrilaterals for a polygonal domain which provides the nodal coordinates and element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is unknown physical variable and in with given Dirichlet boundary conditions over the given convex polygonal domain.

Finite Element Analysis Of Linear Elastic Torsion For Regular Polygons

This paper presents an explicit finite element integration scheme to compute the stiffness matrices for linear convex quadrilaterals. Finite element formulationals express stiffness matrices as double integrals of the products of global derivatives. These integrals can be shown to depend on triple products of the geometric properties matrix and the matrix of integrals containing the rational functions with polynomial numerators and linear denominator in bivariates as integrands over a 2-square. These integrals are computed explicitely by using symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains.We have also developed an automatic all quadrilateral mesh generation technique for convex polygonal domain which provides the nodal coordinates and element connectivity.We have demonstrated the proposed explicit integration scheme to solve the Poisson Boundary Value Problem for a linear elastic torsion of a non-circular bar with cross sections having profiles of equilateral triangle, a square and regular polygons (pentagon(5-gon)to icosagon(20gon)) which are inscribed in a circle of unit radius. Monotonic convergence from below is observed with known analytical solutions for the Prandtl stress function and torsional constant .We have shown the solutions in Tables which list both the FEM and exact solutions. The graphical solutions of contour level curves and the corresponding finite element meshes are also displayed.

TOWARDS A MORE GENERAL TYPE OF UNIVARIATE CONSTRAINED INTERPOLATION WITH FRACTAL SPLINES

Recently, in [Electron. Trans. Numer. Anal. 41 (2014) 420–442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current paper is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.

Environmental Quality Assessment Based on a New Kind of Neural Network Using Rational Spline Weight Functions

As we all known, artificial neural network can be used in the process of environmental quality assessment. To improve the accuracy and science of assessment, a method of environmental quality assessment is presented in this paper, which is based on spline weight function (SWF) neural networks. The weigh functions of the neural network are composed of rational spline functions with cubic numerator and linear denominator (3/1 rational SWF). The simulation results show that, compared with the conventional BP neural networks, this method can get very high precision and accuracy. This case demonstrates that SWF neural networks can offer a very prospective tool for environmental quality assessment.

Sleep Sensor Temperature Prediction Based on Neural Network Training Algorithm with Rational Spline Weight Functions

In wireless sensor network, it is necessary to make effective prediction of sensor node’s data during its sleep period. In this paper a model of rational cubic spline weight function (SWF) neural network with linear denominator was established for sensor node’s temperature prediction. This kind of rational spline function is denoted by 3/1 rational splines. Then we trained and tested the network, the simulation results showed that, compared to the traditional BP neural network, the training speed is higher and the error is smaller. Therefore the prediction model can effectively predict the sensor’s temperature.

Error Analysis of Neural Network with Multiply Neurons Using Rational Spline Weight Functions

In order to describe the generalization ability, this paper discusses the error analysis of neural network with multiply neurons using rational spline weight functions. We use the cubic numerator polynomial and linear denominator polynomial as the rational splines for weight functions. We derive the error formula for approximation, the results can be used to algorithms for training neural networks.

Approximation Analysis for a New Kind of Neural Network Using Rational Spline Weight Functions

To describe the performance for a new kind of neural network, This paper discusses the approximation of the neural network using a kind of rational spline weight function. The rational spline consists of piecewise rational functions with cubic numerators and linear denominators. The theoretic formula of approximation is proposed and an example is also given. It can be concluded that this new neural network can get very high training accuracy.

Computational Complexity of Training Algorithm for a Kind of Neural Network

This paper dose the theoretical analysis for solving the algorithm complexity of a neural network using rational cubic spline weight functions with linear denominator. It concludes that the time complexity of the training algorithm has linear relations with the neural network’s input dimensions, output dimensions as well as the number of samples.

Constrained Control of C^2 Rational Interpolant with Multiple Shape Parameter

A C 2 cubic rational spline with cubic numerator and linear denominator has been constructed .This rational spline belongs to C 2 in the interpolating interval.By selecting the suitable value of shape parameters,it is easy to find the constrains for the shape of interpolating curve to lie above,below or between the given straight lines.Also the error bound for interpolating function is discussed.

Weighted Rational Quartic Spline Interpolation

The weighted rational quartic spline interpolation has been constructed using two kinds of rational quartic spline with linear denominator in this paper. And the conditions of monotonicity preserving and C 2 continuity have been discussed. In addition to, the problem that constrains the weighted interpolating curves to lie above or below a given straight line or quadratic curve has been solved.

A blending interpolator with value control and minimal strain energy

A weighted blending rational interpolator, a kind of smooth interpolator, is constructed using the rational cubic spline with linear denominator and the standard Hermite interpolator. In order to meet the needs of practical design, a new method of value control is employed to control the shape of curves, and the optimal solution for the value control equation with minimal strain energy is derived. The advantage of the method is that it can be applied to modify the local shape of an interpolating curve by selecting suitable parameters and weight coefficients simply. Also when the weight coefficient is in [0,1], the error estimation formula of this interpolator is obtained.

On the denominator values and barycentric weights of rational interpolants

We improve upon the method of Zhu and Zhu [A method for directly finding the denominator values of rational interpolants, J. Comput. Appl. Math. 148 (2002) 341–348] for finding the denominator values of rational interpolants, reducing considerably the number of arithmetical operations required for their computation. In a second stage, we determine the points (if existent) which can be discarded from the rational interpolation problem. Furthermore, when the interpolant has a linear denominator, we obtain a formula for the barycentric weights which is simpler than the one found by Berrut and Mittelmann [Matrices for the direct determination of the barycentric weights of rational interpolation, J. Comput. Appl. Math. 78 (1997) 355–370]. Subsequently, we give a necessary and sufficient condition for the rational interpolant to have a pole.

Error estimation of a kind of rational spline

This paper deals with the approximation properties of a kind of rational spline with linear denominator when the function being interpolated is C 3 in an interpolating interval. Error estimate expressions of interpolating functions are derived, convergence is established, the optimal error coefficient, c i , is proved to be symmetric about the parameters of the rational interpolation and it is bounded. Finally, the precise jump measurements of the second derivatives of the interpolating function at the knots are given.

Error analysis of the derivative of rational interpolation with a linear denominator

This paper deals with the approximation properties of the derivatives of rational cubic interpolation with a linear denominator. Error expressions of the derivatives of interpolating functions are derived, convergence is established and the optimal error coefficient c i is proved to be symmetric about the parameters of the rational interpolation. The unified integral form of the error of the second derivative in all subintervals is obtained. A simple expression of the jump of the second derivative at the knots and the conditions for the interpolating function to be C 2 in the interpolating interval are given.

Constrained control and approximation properties of a rational interpolating curve

This paper deals with the convexity control and the strain energy control of interpolating curves using a rational cubic spline with linear denominator. The sufficient and necessary conditions for controlling the interpolating curve to be convex or concave are derived. When the function being interpolated is f( t)∈ C (3)[ t 0, t n ], the error estimation of the interpolating function and the boundedness of the optimal error coefficient and its double symmetry with regard to parameters are obtained.

Technical Note: Reduction of rational integrals related to linear and convex quadrilateral finite elements

AbstractWe reduce a rational function of bivariate nth degree polynomial numerator with a linear denominator to a simple bivariate polynomial of degree (n − 1) and a rational function of a single variate nth degree polynomial numerator with the same bivariate linear denominator. This has very greatly contributed to the evaluation of (n + 1)(n + 2)/2 rational integrals in bivariates to mere (n + 1) rational integral of a single variate and an integration of simple polynomial in bivariates. Thus the effort of integration is reduced several times and leads to simple analytical expressions in terms of the nodal coordinates. In order to illustrate the numerical process two examples are considered. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 759–770, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10026.