Piecewise Rational Functions Research Articles

Geometric Hermite interpolation by rational curves of constant width

A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve.

A generalized, compactly supported correlation function for data assimilation applications

This work introduces a new, compactly supported correlation function that can be inhomogeneous over Euclidean three‐space, anisotropic when restricted to the sphere, and compactly supported on regions other than spheres of fixed radius. This function, which we call the Generalized Gaspari–Cohn (GenGC) correlation function, is a generalization of the compactly supported, piecewise rational approximation to a Gaussian introduced by Gaspari and Cohn in 1999 and its subsequent extension by Gaspari et al in 2006. The GenGC correlation function is a parametric correlation function that allows two parameters and to vary, as functions, over space, whereas the earlier formulations either keep both and fixed or only allow to vary. Like these earlier formulations, GenGC is a sixth‐order piecewise rational function (fifth‐order near the origin), while the coefficients now depend explicitly on the values of both and at each pair of points being correlated. We show that, by allowing both and to vary, the correlation length of GenGC also varies over space and introduces inhomogeneous and anisotropic features that may be useful in data assimilation applications. Covariances produced using GenGC are computationally tractable due to their compact support and have the added flexibility of generating compact support regions that adapt to the input field. These features can be useful for covariance modeling and covariance tapering applications in data assimilation. We derive the GenGC correlation function using convolutions, discuss continuity properties relating to and and its correlation length, and provide one‐ and two‐dimensional examples that highlight its anisotropy and variable regions of compact support.

Sparse signal reconstruction for nonlinear models via piecewise rational optimization

We propose a method to reconstruct sparse signals degraded by a nonlinear distortion and acquired at a limited sampling rate. Our method formulates the reconstruction problem as a nonconvex minimization of the sum of a data fitting term and a penalization term. In contrast with most previous works which settle for approximated local solutions, we seek for a global solution to the obtained challenging nonconvex problem. Our global approach relies on the so-called Lasserre relaxation of polynomial optimization.We here specifically include in our approach the case of piecewise rational functions, which makes it possible to address a wide class of nonconvex exact and continuous relaxations of the ℓ0 penalization function. Additionally, we study the complexity of the optimization problem. It is shown how to use the structure of the problem to lighten the computational burden efficiently. Finally, numerical simulations illustrate the benefits of our method in terms of both global optimality and signal reconstruction.

Sensor orientation of HaiYang-1C ultraviolet imager based on a piecewise rational function model

Due to the large field angle and large geometric distortions of the HaiYang-1C ultraviolet imager (UVI), a single rational function model (SRFM) is unable to fully absorb the geometric distortions. The SRFM-based sensor orientation accuracy of the UVI images is thereby worse than expected. In order to improve the sensor orientation accuracy, a feasible sensor orientation method for the UVI images based on a piecewise rational function model (PRFM) is proposed. A complete UVI image is first logistically divided into five subimages, and the adjacent subimages have an overlap. Then, an SRFM is used to mathematically fit the physical sensor model of each subimage. Finally, the five SRFMs form a continuous PRFM, and the PRFM-based sensor orientation of the UVI images is performed. Three HaiYang-1C UVI images were tested. The experimental results showed that the unabsorbed geometric distortions fully propagated into the SRFM-based sensor orientation results. The orientation accuracy of the three images reached ∼7 pixels. In the PRFM-based sensor orientation, both the PRFM fitting errors and the inconsistent errors between the adjacent subimages could be negligible. The PRFM-based orientation accuracy was thereby noticeably improved and reached >1 pixel.

Characterizing the value functions of polynomial games

We provide a characterization of the set of real-valued functions that can be the value function of some polynomial game. Specifically, we prove that a function u:R→R is the value function of some polynomial game if and only if u is a continuous piecewise rational function.

Interpolation with spatial rational Pythagorean-hodograph curves of class 4

The paper presents a construction of rational Pythagorean-hodograph curves of class 4 and reveals their properties. In particular, it is shown that each such curve depends on twelve free parameters and has a piecewise rational arc-length function. Geometric interpolation of two data points and two tangent directions is considered in detail and a closed form solution that depends on two shape parameters is given. Regions of shape parameters are derived that imply the interpolant to be regular and admissible. Further, it is shown that one of the shape parameters can be fixed by additionally prescribing also the length of the interpolant. Theoretical results and the construction of G1 Hermite interpolation splines are illustrated with numerical examples.

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.

Solving Kepler’s equation via Smale’s $$\alpha $$ α -theory

We obtain an approximate solution $\tilde{E}=\tilde{E}(e,M)$ of Kepler's equation $E-e\sin(E)=M$ for any $e\in[0,1)$ and $M\in[0,\pi]$. Our solution is guaranteed, via Smale's $\alpha$-theory, to converge to the actual solution $E$ through Newton's method at quadratic speed, i.e. the $n$-th iteration produces a value $E_n$ such that $|E_n-E|\leq (\frac12)^{2^n-1}|\tilde{E}-E|$. The formula provided for $\tilde{E}$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $e=1$ and $M=0$, where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region $[0,1)\times[0,\pi]$ if only rational functions are allowed in each branch.

Positivity-preserving rational bi-cubic spline interpolation for 3D positive data

This paper deals with the shape preserving interpolation problem for visualization of 3D positive data. A required display of 3D data looks smooth and pleasant. A rational bi-cubic function involving six shape parameters is presented for this objective which is an extension of piecewise rational function in the form of cubic/quadratic involving three shape parameters. Simple data dependent constraints for shape parameters are derived to conserve the inherited shape feature (positivity) of 3D data. Remaining shape parameters are left free for designer to modify the shape of positive surface as per industrial needs. The interpolant is not only local, C 1 but also it is a computationally economical in comparison with existing schemes. Several numerical examples are supplied to support the worth of proposed interpolant.

Convex data modelling using rational bi-cubic spline function

A piecewise rational cubic function in the form of a cubic/quadratic has been extended to a rational bicubic blended function to visualize 3D convex data. It involves twelve shape parameters in each rectangular mesh for its representation. Simple data-dependent constraints are imposed on four shape parameters to conserve convexity of convex data. The remaining shape parameters are left up to the user for modification of convex surfaces. Several numerical examples are presented to show the visually pleasing convexity preserving surfaces as compared to existing interpolants.

Shape preserving visualization of monotone data using a rational cubic ball function

This paper deals with the problem of monotonicity-preserving curves of monotonic data. An alternative curve scheme using a piecewise rational cubic ball function is presented. The function involves three shape parameters in each subinterval. Data-dependent constraints are derived for a single shape parameter to preserve the shape of data while the other two are left free to modify the monotonic curve as desired. Several numerical examples are presented to show the effectiveness and capability of the proposed scheme. The scheme is C 2 , flexible, simple, local, and economical.

C1 rational quadratic trigonometric spline

A Bézier like C1 rational quadratic trigonometric polynomial spline is developed. It defines two shape parameters in each subinterval. The approximation and geometric properties are investigated. The curvature continuity is established. The developed rational quadratic trigonometric polynomial spline is extended to C1 piecewise rational bi-quadratic function with four shape parameters in each rectangular patch. Data dependent constraints are developed on the shape parameters in the description of piecewise rational quadratic and bi-quadratic trigonometric polynomial spline for shape preservation of curve and regular surface data. The developed shape preserving schemes provide tangent continuity in quadratic form and does not restrict interval length, derivatives or data.

Modeling rational spline for visualization of shaped data

-The presented paper focuses on two objectives, the development of a new curve interpolation scheme and its application towards the visualization of shaped data. AC1 piecewise rational cubic function, with two parameter family, is developed and presented. The affect of parameters on the shape of the model curve is examined mathematically and illustrated graphically. Simple sufficient data dependent constraints are obtained on one family of the parameters to visualize positive, monotone and convex data. However, the other family of parameters can assume any positive values. The problem of visualization of constrained data is also addressed when the data is lying above a straight line and curve is required to lie on the same side of the line. Moreover, the approximation order of the proposed rational cubic function is also investigated and is found to be O(h3i ).

Fuzzy set theory in modeling uncertainty data via interpolation rational Bezier surface function

Lately, the research of rational spline function widely expanding because the designing of curve and surface, which is the additional parameter known as weight that it had. This weight use to changing the curve and surface shape to fit the real data form. Then, the interpolation method adapted in rational spline function as interpolation rational spline function, which can be constructed by using piecewise rational spline function. This method is being used to model the collection of data set to visualize the data for more the understanding the meaning of the data through the curve especially surface forms. However, the problem arises when the data become uncertain, which is unable to model them. Therefore, in this paper, we introduced fuzzy interpolation rational bicubic Bezier surface (later known as FIRBBS) which modeling the fuzzy data after had been defined by the fuzzy sets definition. The construction of FIRBBS is based on the definition of fuzzy set theory and interpolation rational bicubic Bezier surface. Then, the illustration given in figures for more understanding the method.

Enrichment of a rational polynomial family of shape functions with regularity Ck0, k=0,2,4…

PurposeThe purpose of this paper is to investigate the approximation performance of a family of piecewise rational polynomial shape functions, which are enriched by a set of monomials of order p to obtain high order approximations. To numerically demonstrate the features of the enriched approximation some examples on the mechanical elastic response and free‐vibration of axisymmetric plates and shells are carried out.Design/methodology/approachThe global approximation is based on a particular family of weight function, which is defined on the parametric domain of the element, ξ∈[−1,1], resulting in shape functions with compact support, which have regularity C0k,k=0,2,4… in the global domain Σ. The PU shape functions are enriched by a set of monomials of order p to obtain high order approximation spaces.FindingsBased on the numerical results of elastic axisymmetric plates and shells, it is demonstrated that the proposed methodology produces satisfactory results in terms of keeping the ill‐conditioning of the system of equations under accepted levels. Comparisons are established between linear and Hermitian shape functions showing similar results. The observed results for the free‐vibration problem of plates and shells show the potential of the proposed approximation space.Research limitations/implicationsIn this paper the formulation is limited to the modeling of axisymmetric plate and shell problems. However, it can be applied to model other problems where the high regularity of the approximation is required.Originality/valueThe paper presents an alternative approach to construct partition of unity shape functions based on a particular family of weight function.

Positivity-preserving scattered data interpolation scheme using the side-vertex method

A local surface interpolation scheme has been developed which interpolates and preserves the shape of the positive scattered data arranged over the triangular domain. The proposed interpolating surface has been constructed by using the side vertex method. A C1 piecewise rational cubic function with four parameters has been used for interpolation along each edge of a triangle and along line segments joining a vertex to the opposite edge. Simple sufficient data dependent constraints have been derived on the free parameters in the description of rational cubic function to ensure the shape of positive scattered data.

Shape-preserving curve interpolation

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 is C 3 in an interpolating interval. The order of approximation is .

Positive Data Visualization Using Trigonometric Function

A C1 piecewise rational trigonometric cubic function with four shape parameters has been constructed to address the problem of visualizing positive data. Simple data‐dependent constraints on shape parameters are derived to preserve positivity and assure smoothness. The method is then extended to positive surface data by rational trigonometric bicubic function. The order of approximation of developed interpolant is .

The nearest complex polynomial with a zero in a given complex domain

Given a univariate complex polynomial f and a closed complex domain D , whose boundary C is a curve parameterized by a piecewise rational function, we propose two computational algorithms for finding a univariate complex polynomial f ̃ such that f ̃ has a zero in D and the distance between f and f ̃ is minimal. Our approach is composed of two steps. First, in the case of D consisting of one point α , we give explicit formulas of f ̃ and the minimal distance in terms of α . Next, the case of a general closed domain D is considered by using the property that a nearest polynomial f ̃ has a zero on the boundary C . The curve C is parameterized piecewisely, and on each piece we search for the minimum of the distance between f and f ̃ . At this step we exploit the explicit formula of the minimal distance as a function of a point α . Then the global minimum and the nearest polynomial are obtained by comparing the piecewise minima. Some examples are presented: one of them confirms that the distance between a nearest complex polynomial and a given polynomial is less than that between a nearest real polynomial and the given polynomial.

Computing the nearest polynomial with a zero in a given domain by using piecewise rational functions

For a real univariate polynomial f and a closed complex domain D whose boundary C is a simple curve parameterized by a univariate piecewise rational function, a rigorous method is given for finding a real univariate polynomial f ̃ such that f ̃ has a zero in D and ‖ f − f ̃ ‖ ∞ is minimal. First, it is proved that the minimum distance between f and polynomials having a zero at α ∈ C is a piecewise rational function of the real and imaginary parts of α . Thus, on C , the minimum distance is a piecewise rational function of a parameter obtained through the parameterization of C . Therefore, f ̃ can be constructed by using the property that f ̃ has a zero on C and computing the minimum distance on C . We analyze the asymptotic bit complexity of the method and show that it is of polynomial order in the size of the input.