Polynomial Weight Functions Research Articles

On a Sixth-order Solver for Multiple Roots of Nonlinear Equations

A number of iterative schemes with high convergence order to solve nonlinear equations are presented in the literature. In this paper, a sixth-order multiple-zero finder has been developed and the dynamics of selected iterative schemes with uniparametric polynomial weight function are investigated using M\"{o}bius conjugacy map applied to the form $((z-A)(z-B))^m$. The complex dynamics on the Riemann sphere by analyzing the parameter spaces associated with the free critical points are studied and the numerical experiments are carried out.

Frequency measurement research with weight averaging of pulse output signal of voltage-to-frequency converter

The paper presents the essence and investigation of the efficiency of weight averaging of a pulse output signal of voltage-to-frequency converter. The effect of counting and the influence of interference on the result of weight averaging of frequency modulated pulses are analyzed. It is shown that from the point of view of counting error reduction, the best are polynomial weight functions. In the case of high interferences whose frequencies are unstable or may change over a wide range, the specified level of their suppression, together with the reduction of the counting effect, can be achieved using weight functions, such as trigonometric. The interference suppression both in the narrow and wide frequency ranges, as well as the counting error when using selected weight functions, have been tested by both simulation and experimentally. The obtained results show very good convergence with the values calculated from theoretical formulas.

Scalar and vector tomography for the weighted transport equation with application to helioseismology

Abstract Motivated by the application to helioseismology, we demonstrate uniqueness and stability for a class of inverse problems of the weighted transport equation. Using 𝐴-analytic functions, this inverse problem is expressed as a Cauchy problem. In this form, we show that, for a finite even trigonometric polynomial weight function, the resulting system is well-conditioned numerically and permits a Carleman-like estimate with boundary terms.

Transmission dynamic of stochastic hepatitis C model by spectral collocation method

In this research work, we present an exciting mathematical analysis of a stochastic model, using a standard incidence function, for infectious disease hepatitis C transmission dynamics. In this model, we divided the infected population into three different classes with two different infection stages known as chronic class and acute class while the third is an isolation class. We also presents briefly the Legendre spectral method for the numerical solution to the proposed model. It is observed that the disease-free equilibrium is asymptotically stable, when basic reproduction number It is also shown that the proposed model has a stable endemic equilibrium when the reproduction number Also, sensitivity analysis is carried out to study and identify the effect of parameters on R 0. Moreover, we have performed numerical simulations to study the influence of disease free equilibrium and endemic equilibrium. Legendre polynomial and Legendre weight function are used to solve the proposed stochastic system numerically. Numerical results are compared against the basic reproduction number.

A General Class of [formula omitted] Smooth Rational Splines: Application to Construction of Exact Ellipses and Ellipsoids

In this paper, we describe a general class of C1 smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids — some of the most important primitives for CAD and CAE. The univariate rational splines are assembled by transforming multiple sets of NURBS basis functions via so-called design-through-analysis compatible extraction matrices; different sets of NURBS are allowed to have different polynomial degrees and weight functions. Tensor products of the univariate splines yield multivariate splines. In the bivariate setting, we describe how similar design-through-analysis compatible transformations of the tensor-product splines enable the construction of smooth surfaces containing one or two polar singularities. The material is self-contained, and is presented such that all tools can be easily implemented by CAD or CAE practitioners within existing software that support NURBS. To this end, we explicitly present the matrices (a) that describe our splines in terms of NURBS, and (b) that help refine the splines by performing (local) degree elevation and knot insertion. Finally, all C1 spline constructions yield spline basis functions that are locally supported and form a convex partition of unity.

Weighted Hurwitz Numbers and Topological Recursion

The KP and 2D Toda \(\tau \)-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel–Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations in the case of polynomial weight functions.

Optimization for maximizing the impact-resistance of patch repaired CFRP laminates using a surrogate-based model

This study proposes a surrogate-based optimization model of the external patch design for damaged carbon fiber reinforced polymer (CFRP) laminates to improve the impact-resistance. Initially, finite element (FE) models of patch-repaired CFRP laminates constructed using continuum damage mechanics (CDM) approach and cohesive zone model (CZM), those well capture the intra- and inter-laminar damages during the low-velocity impacts, respectively. Experimental measurements of low-velocity impact tests concur with the numerical predictions and validate the FE models. Subsequently, the validated models are used to optimize the design parameters of repair patch, consisting of patch radius, thickness and off-axis angle. The optimum design parameters are identified using the surrogate models which are constructed using Diffuse Approximation and Design of Experiments (DOE). The identified optimum patch configurations significantly enhance the impact-resistance of repaired CFRP laminates by decreasing the impact energy absorption and the delamination surface area. Finally, the robustness of the optimization model is confirmed via error analysis of surrogate model reconstruction, and testing the influence of polynomial basis and weighting functions.

Exact and Approximate Weighted Model Integration with Probability Density Functions Using Knowledge Compilation

Weighted model counting has recently been extended to weighted model integration, which can be used to solve hybrid probabilistic reasoning problems. Such problems involve both discrete and continuous probability distributions. We show how standard knowledge compilation techniques (to SDDs and d-DNNFs) apply to weighted model integration, and use it in two novel solvers, one exact and one approximate solver. Furthermore, we extend the class of employable weight functions to actual probability density functions instead of mere polynomial weight functions.

Robust forecasting of electricity prices: Simulations, models and the impact of renewable sources

In this paper a robust approach to modeling electricity spot prices is introduced. Differently from what has been recently done in the literature on electricity price forecasting, where the attention has been mainly drawn by the prediction of spikes, the focus of this contribution is on the robust estimation of nonlinear SETARX models (Self-Exciting Threshold Auto Regressive models with eXogenous regressors). In this way, parameter estimates are not, or very lightly, influenced by the presence of extreme observations and the large majority of prices, which are not spikes, could be better forecasted. A Monte Carlo study is carried out in order to select the best weighting function for Generalized M-estimators of SETAR processes. A robust procedure to select and estimate nonlinear processes for electricity prices is introduced, including robust tests for stationarity and nonlinearity and robust information criteria. The application of the procedure to the Italian electricity market reveals the forecasting superiority of the robust GM-estimator based on the polynomial weighting function with respect to the non-robust Least Squares estimator. Finally, the introduction of generation from renewable sources in the robust estimation of SETARX processes contributes to the improvement of the forecasting ability of the model.

Developing an Optimal Class of Generic Sixteenth-Order Simple-Root Finders and Investigating Their Dynamics

Developed here are sixteenth-order simple-root-finding optimal methods with generic weight functions. Their numerical and dynamical aspects are investigated with the establishment of a main theorem describing the desired optimal convergence. Special cases with polynomial and rational weight functions have been extensively studied for applications to real-world problems. A number of computational experiments clearly support the underlying theory on the local convergence of the proposed methods. In addition, to investigate the relevant global convergence, we focus on the dynamics of the developed methods, as well as other known methods through the visual description of attraction basins. Finally, we summarized the results, discussion, conclusion, and future work.

A study of dynamics via Möbius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions

A generic family of sixth-order modified Newton-like multiple-zero finders have been proposed in Geum et al. (2016). Among them we select a specific family of iterative methods with uniparametric bivariate polynomial weight functions and study their dynamics by investigating the relevant parameter spaces and dynamical planes, via Möbius conjugacy map applied to a prototype polynomial of the form ( z − a ) m ( z − b ) m . The resulting dynamics is best illustrated through a variety of parameter spaces as well as dynamical planes.

The Influence of Renewables on Electricity Price Forecasting: A Robust Approach

In this paper a robust approach to modelling electricity spot prices is introduced. Differently from what has been recently done in the literature on electricity price forecasting, where the attention has been mainly drawn by the prediction of spikes, the focus of this contribution is on the robust estimation of nonlinear SETARX models (Self-Exciting Threshold Auto Regressive models with eXogenous regressors). In this way, parameters estimates are not, or very lightly, influenced by the presence of extreme observations and the large majority of prices, which are not spikes, could be better forecasted. A Monte Carlo study is carried out in order to select the best weighting function for Generalized M-estimators of SETAR processes. A robust procedure to select and estimate nonlinear processes for electricity prices is introduced, including robust tests for stationarity and nonlinearity and robust information criteria. The application of the procedure to the Italian electricity market reveals the forecasting superiority of the robust GM-estimator based on the polynomial weighting function respect to the non-robust Least Squares estimator. Finally, the introduction of external regressors in the robust estimation of SETARX processes contributes to the improvement of the forecasting ability of the model.

Properties of interpolatory quadrature with equidistant nodes on the unit circle

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error.

A discontinuous Petrov–Galerkin (DPG) formulation for the FEM applied to the Helmholtz equation for high wavenumbers

Pollution error is a well known source of inaccuracies in continuous or discontinuous FE approaches to solve the Helmholtz equation. This topic is widely studied in a large number of papers, e.g., (Ihlenburg and Babuska, Comput Math Appl 30(9):9–37, 1995; Ihlenburg, Finite element analysis of acoustic scattering applied mathematical sciences, vol 132. Springer, New York, 1998). Robust methodologies for structured square meshes have been developed in recent years. This work seeks to develop a methodology based on Petrov–Galerkin discontinuous formulation, to minimize phase error for Helmholtz equation for both structured and unstructured meshes. A Petrov–Galerkin finite element formulation is introduced for the Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the triangular mesh, a global basis function for the weighting space is obtained, adding bilinear $$C^0$$ Lagrangian weighting function linear combinations. The optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients of these linear combinations attending to optimal criteria. This is done numerically through a preprocessing technique that is naturally applied to non-uniform and unstructured meshes. In particular, for uniform meshes a quasi optimal interior stencil of the same order of the quasi-stabilized finite element method stencil derived by (Babuska et al., Comput Meth Appl Mech Eng 128:325–359, 1995) is obtained. The numerical results indicate a better performance in relation to the classic discontinuous Galerkin method.

Stress intensity factors for embedded elliptical cracks in cylindrical and spherical vessels

In this paper, we give a very accurate approximation of the stress intensity factors of embedded elliptical cracks in cylindrical and spherical vessels. We evaluate the stress intensity factor along the whole crack border; we do so using a polynomial weight function based on a second order approximation of the Oore-Burns integral in terms of the deviation of the contour from a disk. The stress intensity factor is given for uniform internal pressure and is related to the hoop stress. Finally, a comparison with FE stress intensity factor is given.

A Cellular Network Architecture With Polynomial Weight Functions

Emulations of cellular nonlinear networks on digital reconfigurable hardware are renowned for an efficient computation of massive data, exceeding the accuracy and flexibility of full-custom designs. In this contribution, a digital implementation with polynomial coupling weight functions is proposed for the first time, establishing novel fields of application, e.g., in the medical signal processing and in the solution of partial differential equations. We present an architecture that is capable of processing large-scale networks with a high degree of parallelism, implemented on state-of-the-art field-programmable gate arrays.

Training Neural Networks by Continued Fractional Weight Functions

To solve the training problems for complex problems, a new learning algorithm based on continued fractional weight function for training neural networks is proposed, which is different from that of polynomial weight functions. The continued fractional weight functions using interpolation methods are more suitable for training the patterns obtained by some rational problems. The analysis of generalization is also presented in this paper. At last, to illustrate the power of the new learning algorithm, a simulation example is presented to show that the new algorithm proposed in this paper has good performance both on generalization and calculating precision.

An Optimal Family of Fast 16th-Order Derivative-Free Multipoint Simple-Root Finders for Nonlinear Equations

This paper investigates an optimal family of derivative-free fast 16th-order multipoint iterative methods for solving nonlinear equations using polynomial weighting functions and a real control parameter. Convergence analyses and computational properties are shown along with a comparison of the classical work done by Kung–Traub in 1974. The underlying theoretical treatment and computational advantage of faster computing time is well supported through a variety of concrete numerical examples.

Optimal Designs for Rational Function Regression

We consider the problem of finding optimal nonsequential designs for a large class of regression models involving polynomials and rational functions with heteroscedastic noise also given by a polynomial or rational weight function. Since the design weights can be found easily by existing methods once the support is known, we concentrate on determining the support of the optimal design. The proposed method treats D-, E-, A-, and Φ p -optimal designs in a unified manner, and generates a polynomial whose zeros are the support points of the optimal approximate design, generalizing a number of previously known results of the same flavor. The method is based on a mathematical optimization model that can incorporate various criteria of optimality and can be solved efficiently by well-established numerical optimization methods. In contrast to optimization-based methods previously proposed for the solution of similar design problems, our method also has theoretical guarantee of its algorithmic efficiency; in concordance with the theory, the actual running times of all numerical examples considered in the paper are negligible. The numerical stability of the method is demonstrated in an example involving high-degree polynomials. As a corollary, an upper bound on the size of the support set of the minimally supported optimal designs is also found.

Comparison of local weak and strong form meshless methods for 2-D diffusion equation

A comparison between weak form meshless local Petrov–Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.