Newton Polynomial Research Articles

Optimal interval length for the collocation of the Newton interpolation basis

It is known that the Lagrange interpolation problem at equidistant nodes is ill-conditioned. We explore the influence of the interval length in the computation of divided differences of the Newton interpolation formula. Condition numbers are computed for lower triangular matrices associated to the Newton interpolation formula at equidistant nodes. We consider the collocation matrices L and PL of the monic Newton basis and a normalized Newton basis, so that PL is the lower triangular Pascal matrix. In contrast to L, PL does not depend on the interval length, and we show that the Skeel condition number of the (n + 1) × (n + 1) lower triangular Pascal matrix is 3n. The ∞-norm condition number of the collocation matrix L of the monic Newton basis is computed in terms of the interval length. The minimum asymptotic growth rate is achieved for intervals of length 3.

Conversions between barycentric, RKFUN, and Newton representations of rational interpolants

We derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm Y. Nakatsukasa et al. (2018) [6] within the Rational Krylov Toolbox and for the solution of nonlinear eigenvalue problems.

Error-Controlled Model Approximation for Gaussian Process Morphable Models

Gaussian Process Morphable Models (GPMMs) unify a variety of non-rigid deformation models for surface and image registration. Deformation models, such as B-splines, radial basis functions, and PCA models are defined as a probability distribution using a Gaussian process. The method depends heavily on the low-rank approximation of the Gaussian process, which is mandatory to obtain a parametric representation of the model. In this article, we propose the use of the pivoted Cholesky decomposition for this task, which has the following advantages: (1) Compared to the current state of the art used in GPMMs, it provides a fully controllable approximation error. The algorithm greedily computes new basis functions until the user-defined approximation accuracy is reached. (2) Unlike the currently used approach, this method can be used in a black-box-like scenario, whereas the method automatically chooses the amount of basis functions for a given model and accuracy. (3) We propose the Newton basis as an alternative basis for GPMMs. The proposed basis does not need an SVD computation and can be iteratively refined. We show that the proposed basis functions achieve competitive registration results while providing the mentioned advantages for its computation.

Linearizations of matrix polynomials in Newton bases

We discuss matrix polynomials expressed in a Newton basis, and the associated polynomial eigenvalue problems. Properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces. Also, we show how the family of Fiedler pencils can be adapted to matrix polynomials expressed in a Newton basis. These new Newton–Fiedler pencils are shown to be strong linearizations, and some computational aspects related to them are discussed.

Local variably scaled Newton basis functions collocation method for solving Burgers’ equation

In the present paper, a meshless method which constructs derivatives discretizations based on variably scaled Newton basis functions (VSNBFs) interpolants for localized sets of nodes is provided to solve Burgers’ equation in the “native” Hilbert space of the reproducing kernel. Using such formulas for discretizations of PDEs when each internal node considered as a center, leads to a sparse global of system of ordinary differential equations (ODEs) to be formed by a simple assembly of function values at the local system centrepoints. For efficiency and stability reasons, we use the interpolation with variably scaled kernels, which are introduced recently by Bozzini et al. The VSNBFs can be constructed from a symmetric positive definite variably scaled kernel. The main advantage of using the local method is that the overlapping domains of influence result in many small matrices with the dimension of the number of nodes included in the domain of influence for each center node, instead of a large collocation matrix, and hence, sparse global derivative matrices from local contributions. Therefore, the method needs less computer storage and flops. The method is tested on six benchmark problems on Burgers’ equation with mixed boundary conditions. Five number of nodes are used in the domain of influence of the local support. The numerical results for different values of Reynolds numbers (Re) are compared with analytical solutions as well as some other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re. Numerical results also show that using variably scaled kernels leads to results that are better than using constant scaled kernels, with respect to both stability and error.

A localized Newton basis functions meshless method for the numerical solution of the 2D nonlinear coupled Burgers’ equations

PurposeThe purpose of this paper is to introduce a local Newton basis functions collocation method for solving the 2D nonlinear coupled Burgers’ equations. It needs less computer storage and flops than the usual global radial basis functions collocation method and also stabilizes the numerical solutions of the convection-dominated equations by using the Newton basis functions.Design/methodology/approachA meshless method based on spatial trial space spanned by the local Newton basis functions in the “native” Hilbert space of the reproducing kernel is presented. With the selected local sub-clusters of domain nodes, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs).FindingsThe method is successfully used for solving the 2D nonlinear coupled Burgers’ equations for reasonably high values of Reynolds number (Re). It is a well-known issue in the analysis of the convection-diffusion problems that the solution becomes oscillatory when the problem becomes convection-dominated if the standard methods are followed without special treatments. In the proposed method, the authors do not detect any instability near the front, hence no technique is needed. The numerical results show that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re.Originality/valueThe authors used more stable basis functions than the standard basis of translated kernels for representing of kernel-based approximants for the numerical solution of partial differential equations (PDEs). The local character of the method, having a well-structured implementation including enforcing the Dirichlet and Neuman boundary conditions, and producing accurate and stable results for flow with reasonably high values of Re for the numerical solution of the 2D nonlinear coupled Burgers’ equations without any special technique are the main values of the paper.

Syzygies over the polytope semiring

Tropical geometry and its applications indicate a theory of over polytope semirings. Taking cue from this indication, we study a notion of syzygies over the polytope semiring. We begin our exploration with the concept of Newton basis, an analogue of Grobner basis that captures the image of an ideal under the Newton polytope map. The image ${\rm New}(I)$ of a graded ideal $I$ under the Newton polytope is a graded sub-semimodule of the polytope semiring. Analogous to the Hilbert series, we define the notion of Newton-Hilbert series that encodes the rank of each graded piece of ${\rm New}(I)$. We prove the rationality of the Newton-Hilbert series for sub-semimodules that satisfy a property analogous to Cohen-Macaulayness. We define notions of regular sequence of polytopes and syzygies of polytopes. We show an analogue of the Koszul property characterizing the syzygies of a regular sequence of polytopes.

Polynomial interpolation on interlacingrectangular grids

In this paper we establish the unisolvence of any interlacing pair of rectangular grids of points with respect to a large class of associated polynomial spaces. This includes interpolation on Padua points and the schemes of Morrow and Patterson (1978), Xu (1996), and Erb et al. (2016). We use Newton polynomials and divided differences and an apparently new formula for determinants of certain divided difference matrices.

Using Newton basis functions for solving diffusion equation with nonlocal boundary condition

In this paper we introduce a meshless method based on spatial trial spaces spanned by the Newton basis functions. The Newton basis arising from a pivoted Cholesky factorization turns out to be stable and computationally cheap while being orthonormal in the “native” Hilbert space of the kernel. This method can overcome on some of challenges of the full meshless kernel-based methods such as ill-conditioning associated with the dense linear systems that arise. This approach is used to obtain the numerical solution of time-dependent partial differential equations (PDEs), diffusion equation with two types of nonlocal boundary condition. The time-dependent PDE is discretized in space by the collocation method based on Newton basis functions as trial functions and it is converted to a system of ordinary differential equations in time. Stability of the scheme is discussed. Numerical examples are provided to show the accuracy and easy implementation of the novel proposed method.

A Simple Model for Solvent Evaporation in Electrospinning Process

One of the most important issues in electrospinning process is whipping instability and solvent evaporation that plays a critical role in nanofiber formation. This report simulates whipping process using numerical methods for a bead-viscoelastic element fiber model combined with a polynomial equation for adding solvent evaporation influence on the three-dimensional path. The suggested equation has been obtained by applying Newton interpolation polynomial operation. Then the results of nanofiber diameters for experimental data, original model and modified model are compared with different statistical methods. ANOVA analysis shows the prediction of diameters could be more accurate and meaningful. Meanwhile, a calculation forecast accuracy chart is presented in detail. It is observed that our suggested model shows more accuracy and is more reliable than the bead-spring model.

Propagating properties of spatial solitons in the competing nonlocal cubic-quintic nonlinear media

We study the new spatial optical solitons and their propagating properties in the one-dimensional nonlocal cubic-quintic (C-Q) nonlinear model by the numerical method. We obtain multi-bright solitons and multipole soliton solutions in the one-dimensional nonlocal C-Q nonlinear model. The propagation of bright solitons is stable in the competing nonlocal cubic self-defocusing and quintic self-focusing nonlinear media when these nonlocal and nonlinear parameters are in the appropriate value domain. Considering the different nonlinear cubic effects, the interaction between two optical solitons with the same phase in the general nonlocal media displays the attraction or the repulsion for different nonlocal and nonlinear parameters. We find that the interval of two solitons affects the interaction between them. The refractive index is changed with the propagating constant when the nonlocal constant d3 is 10. Moreover, the triplepole, quadrupole and pentapole solitons can propagate steadily when the nonlocal parameters are appropriate, but hexa-pole (or above) solitons propagate unsteadily for any nonlocal parameter. Furthermore, we investigate the multi-pole solitons and their propagation stabilities by the Newton difference method and the Fourier split step method, obtain the stable propagation conditions for dipole, triplepole and quadrupole solitons, and find that the propagation of the pentapole and higher-order pole solitons is unstable. We also discuss the interactions of multi-pole solitons when they propagate along the axis z. The interactions are attraction or repulsion when the nonlocal and the nonlinear parameters are different. Meanwhile, we simulate the evolution of the refractive index along the axis z when the spatial optical solitons are multi-pole solitons. Finally, we study the relation between the power of soliton and the propagation constant under different degree of nonlocality. The power of the single bright soliton does not monotonically increase with the increasing propagation constant when the degree of nonlocality d3 is 10. We also derive the relation between the power of dipole bright solitons with the cubic nonlinearity parameter and the propagation constant under different degree of nonlocality. The power decreases monotonically with the increasing propagation constant when the cubic nonlinearity is a certain value or with the increasing cubic nonlinearity when the propagation constant is a certain value.

A local reproducing kernel method accompanied by some different edge improvement techniques: application to the Burgers’ equation

The paper introduces a local reproducing kernel method based on spatial trial space spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. It is a truly meshless approach which uses the local sub clusters of domain nodes, named local domains of influence, for the calculation of the spatial partial derivatives. With the selected domain of influence, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. After the successful approximation function creation, all the needed differential operators can be constructed by applying an arbitrary operator on the approximation function. The main advantage of using the local method is that the overlapping domains of influence results in many small matrices with the dimension of the number of nodes included in the domain of influence for each center node, instead of a large collocation matrix, and hence the sparse global derivative matrices from local contributions. So the method needs less computer storage and flops. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs). This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due to their suitability and better accuracy. To decrease accuracy degradations near boundaries, some different edge improvement techniques are also used. The method is tested on four benchmark problems on Burgers’ equation with mixed boundary conditions. Five nodded domains of influence are used in the local support. The numerical results for different value of Reynolds numbers (Re) is compared with analytical solution as well as some other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re.

Linear interpolation for spatial data grid by newton polynomials

Linear interpolation for spatial data grid by newton polynomials

Operator Newton polynomials and well-solvable problems for generalized Euler equation

The study of well-solvable operator equations in a Banach space, which was initiated by the authors in [4, 5], is continued. Namely, it is proved by means of Maslov’s operator method that a polynomial equation with abstract Newton polynomials is well solvable in the sense of Hadamard. The obtained results are applied to prove that a large class of problems for differential equations with variable coefficient having a singularity (such equations are called generalized Euler equations in the paper) are well solvable.

Signed polyomino tilings by n-in-line polyominoes and Gröbner bases

Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits a signed tiling by three-in-line polyominoes (tribones) if and only if m 2 {9d?1, 9d}d2N. We apply the theory of Gr?bner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m 2 {dn2 ? 1, dn2}d2N. Explicit description of the Gr?bner basis allows us to calculate the ?Gr?bner discrete volume? of a lattice region by applying the division algorithm to its ?Newton polynomial?. Among immediate consequences is a description of the tile homology group for the n-in-line polyomino.

Numerically stable formulas for a particle-based explicit exponential integrator

Numerically stable formulas are presented for the closed-form analytical solution of the X-IVAS scheme in 3D. This scheme is a state-of-the-art particle-based explicit exponential integrator developed for the particle finite element method. Algebraically, this scheme involves two steps: (1) the solution of tangent curves for piecewise linear vector fields defined on simplicial meshes and (2) the solution of line integrals of piecewise linear vector-valued functions along these tangent curves. Hence, the stable formulas presented here have general applicability, e.g. exact integration of trajectories in particle-based (Lagrangian-type) methods, flow visualization and computer graphics. The Newton form of the polynomial interpolation definition is used to express exponential functions of matrices which appear in the analytical solution of the X-IVAS scheme. The divided difference coefficients in these expressions are defined in a piecewise manner, i.e. in a prescribed neighbourhood of removable singularities their series approximations are computed. An optimal series approximation of divided differences is presented which plays a critical role in this methodology. At least ten significant decimal digits in the formula computations are guaranteed to be exact using double-precision floating-point arithmetic. The worst case scenarios occur in the neighbourhood of removable singularities found in fourth-order divided differences of the exponential function.

A Comparative Analysis of Methods of Polynomial Coefficients Determination for Reliability Indices of Ilorin Distribution Systems

Distribution system delivers the transmitted energy to the end users which are customers. This paper develops a generalized polynomial model and compares the coefficients for the assessment of reliability indices of Ilorin distribution system feeders. The results of the coefficients obtained were validated by subjecting the polynomial model to three methods namely: Lagrange Polynomial, Newton and Chebychev Polynomial function methods. The results of the work showed that average values of -7x10 -4 , 0.00212, -0.0256, 0.1549, - 0.4865 and 0.7049 were obtained as the coefficient values from the Lagrange, Newton and Chebychev polynomial function methods which compare favourably well with the coefficient values of -7x10 -4 , 0.0021, -0.0257, 0.1549, -0.4865 and 0.7048 obtained from the simulation results. The

A Galerkin-reproducing kernel method: Application to the 2D nonlinear coupled Burgers' equations

Abstract The paper introduces a Galerkin method in the reproducing kernel Hilbert space. It is implemented as a meshless method based on spatial trial spaces spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. For the time-dependent PDEs it leads to a system of ordinary differential equations. The method is used for solving the 2D nonlinear coupled Burgers' equations having Dirichlet and mixed boundary conditions. The numerical solutions for different values of Reynolds number (Re) are compared with analytical solutions as well as other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Re in the case of Dirichlet boundary conditions.

The computer method of variable piecewise polynomial approximation of functions and solutions of ordinary differential equations

The authors present a computer method of piecewise polynomial approximation of functions and solution of Cauchy problem for systems of ordinary differential equations based on the Newton polynomial. The approximating polynomial on a subinterval is converted to the form with numerical coefficients, the degree of the polynomial and the number of subintervals varies. The method is shown to uniformly converge at the rate of geometric progression under conditions of double continuous differentiability of the function and of the right-hand side of the system. The approximate solution of the system is continuous, continuously differentiable, and is characterized by low error rate, in particular, when solving stiff problems.

Interpolación de la probabilidad de ruina con reclamaciones tipo fase, usando polinomios de Newton.

En este artículo se presenta la manera de obtener fórmulas explícitas bajo condiciones especiales o por medio de polinomios interpoladores de Newton para aproximar la Probabilidad de Ruina en el caso de riesgo a tiempo continuo en el marco de un Modelo de Riesgo Colectivo cuando los montos de las reclamaciones tienen una distribución de Tipo Fase.