Determining the multiplicity of a root of a nonlinear algebraic equation

Newton’s method is most frequently used to find the roots of a nonlinear algebraic equation. The convergence domain of Newton’s method can be expanded by applying a generalization known as the continuous analogue of Newton’s method. For the classical and generalized Newton methods, an effective root-finding technique is proposed that simultaneously determines root multiplicity. Roots of high multiplicity (up to 10) can be calculated with a small error. The technique is illustrated using numerical examples.     

Efficient solution techniques for implicit finite element schemes with flux limiters

The algebraic flux correction (AFC) paradigm is equipped with efficient solution strategies for implicit time‐stepping schemes. It is shown that Newton‐like techniques can be applied to the nonlinear systems of equations resulting from the application of high‐resolution flux limiting schemes. To this end, the Jacobian matrix is approximated by means of first‐ or second‐order finite differences. The edge‐based formulation of AFC schemes can be exploited to devise an efficient assembly procedure for the Jacobian. Each matrix entry is constructed from a differential and an average contribution edge by edge. The perturbation of solution values affects the nodal correction factors at neighbouring vertices so that the stencil for each individual node needs to be extended. Two alternative strategies for constructing the corresponding sparsity pattern of the resulting Jacobian are proposed. For nonlinear governing equations, the contribution to the Newton matrix which is associated with the discrete transport operator is approximated by means of divided differences and assembled edge by edge. Numerical examples for both linear and nonlinear benchmark problems are presented to illustrate the superiority of Newton methods as compared to the standard defect correction approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonequilibrium temperatures and second-sound propagation along nanowires and thin layers

It is shown that the dispersion relation of heat waves along nanowires or thin layers could allow to compare two different definitions of nonequilibrium temperature, since thermal waves are predicted to propagate with different phase speed depending on the definition of nonequilibrium temperature being used. The difference is small, but it could be in principle measurable in nanosystems, as for instance nanowires and thin layers, in a given frequency range. Such an experiment could provide a deeper view on the problem of the definition of temperature in nonequilibrium situations.     

A Newton iterative mixed finite element method for stationary conduction–convection problems

In this article, a Newton iterative mixed finite element method is presented for solving the stationary conduction–convection problems in two dimensions. The stability and the errors generated by both partitioning the space and solving nonlinear equations are analysed, which show that our method is stable and has good precision. Finally, some numerical experiments are given to confirm its effect.     

Numerical Approximation of a Nonlinear 3D Heat Radiation Problem

In this paper, we are concerned with the numerical approximation of a steady-state heat radiation problem with a nonlinear Stefan-Boltzmann boundary condition in$\mathbb{R}^3$. We first derive an equivalent minimization problem and then present a finite element analysis to the solution of such a minimization problem. Moreover, we apply the Newton iterative method for solving the nonlinear equation resulting from the minimization problem. A numerical example is given to illustrate theoretical results.     

A smoothing Newton method for mathematical programs constrained by parameterized quasi-variational inequalities

We consider a class of mathematical programs governed by parameterized quasi-variational inequalities(QVI).The necessary optimality conditions for the optimization problem with QVI constraints are reformulated as a system of nonsmooth equations under the linear independence constraint qualification and the strict slackness condition.A set of second order sufficient conditions for the mathematical program with parameterized QVI constraints are proposed,which are demonstrated to be sufficient for the second o...     

Sixth order derivative free family of iterative methods

In this study, we develop a four-parameter family of sixth order convergent iterative methods for solving nonlinear scalar equations. Methods of the family require evaluation of four functions per iteration. These methods are totally free of derivatives. Convergence analysis shows that the family is sixth order convergent, which is also verified through the numerical work. Though the methods are independent of derivatives, computational results demonstrate that family of methods are efficient and demonstrate equal or better performance as compared with other six order methods, and the classical Newton method.     

Singularly Perturbed Nonlinear Integrodifferential Systems with Fast Varying Kernels

We consider nonlinear singularly perturbed integro-differential equations with fast varying kernels. It is assumed that the spectrum of the limiting operator lies in the closed left half-plane Re0. We derive an algorithm for obtaining regularized (in the sense of Lomov) asymptotic solutions in both the nonresonance and resonance cases. In deriving the algorithm, we essentially use the regularization apparatus for integral operators with fast varying kernels, developed earlier by the authors for linear integral and integro-differential systems. The algorithm is justified and the existence of a solution of the original nonlinear problem is proved by means of the Newton method for operator equations.     

Simultaneous Point Estimates for Newton's Method

Beside the classical Kantorovich theory there exist convergence criteria for the Newton iteration which only involve data at one point, i.e. point estimates. Given a polynomial P, these conditions imply the point evaluation of n = deg(P) functions (from a certain Taylor expansion). Such sufficient conditions ensure quadratic convergence to a single zero and have been used by several authors in the design and analysis of robust, fast and efficient root-finding methods for polynomials.In this paper a sufficient condition for the simultaneous convergence of the one-dimensional Newton iteration for polynomials will be given. The new condition involves only n point evaluations of the Newton correction and the minimum mutual distance of approximations to ensure simultaneous quadratic convergence to the pairwise distinct n roots.     

A new approach to vector-valued rational interpolation

In this work we propose three different procedures for vector-valued rational interpolation of a function F(z), where , and develop algorithms for constructing the resulting rational functions. We show that these procedures also cover the general case in which some or all points of interpolation coalesce. In particular, we show that, when all the points of interpolation collapse to the same point, the procedures reduce to those presented and analyzed in an earlier paper (J. Approx. Theory 77 (1994) 89) by the author, for vector-valued rational approximations from Maclaurin series of F(z). Determinant representations for the relevant interpolants are also derived.