Some remarks concerning the convergence of the numerical-analytic method of successive approximations

We establish new improved estimates necessary for the justification of the numerical-analytic method for the investigation of the existence and construction of approximate solutions of nonlinear boundary-value problems for ordinary differential equations.     

Justification of a numerical-analytic method of successive approximations for problems with integral boundary conditions

A justification is presented for application of a numerical-analytic method of successive approximations to investigation and approximate construction of solutions of differential equations with integral boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1231–1239, September, 1991.     

The successive approximations method for the airfoil equation

The successive approximations (or Neumann iterations) method for the solution of Fredholm integral equations of the second kind is applied here for the first time, after an appropriate modification, to a Cauchy-type singular integral equation of the first kind, the airfoil equation. The convergence of the method is investigated and three simple applications are made. The numerical implementation of the method (by using Gaussian quadrature rules) is also described in detail and numerical results verifying the accuracy and convergence of the method are displayed.     

Global convergence of the method of successive approximations on S1

The class of iterating functions of C(S¹, S¹) for which the method of successive approximations converges for any starting point is characterized; such characterization is given by (i) the existence of a fixed point; (ii) the non-existence of periodic points of an even period.     

Global convergence of the method of successive approximations on S1

The class of iterating functions of C(S¹, S¹) for which the method of successive approximations converges for any starting point is characterized; such characterization is given by (i) the existence of a fixed point; (ii) the non-existence of periodic points of an even period.     

A method of successive approximations for nonlinear viscoelasticity based on the nonlinear correspondence principle and the method of approximations

Mechanics of Composite Materials -     

Using successive approximations for improving the convergence of GMRES method

In this paper, our attention is concentrated on the GMRES method for the solution of the system (I–T)x=b of linear algebraic equations with a nonsymmetric matrix. We perform m pre-iterations y _l+1 =T _yl +b before starting GMRES and put y _m for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the mth powers of eigenvalues of the matrix T Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical experiments verify that it is advisable to perform pre-iterations before starting GMRES as they require fewer arithmetic operations than GMRES. Towards the end of the paper we present a numerical experiment for a system obtained by the finite difference approximation of convection-diffusion equations.     

A comparison between the variational iteration method and the successive approximations method

In this paper we investigate and compare the variational iteration method and the successive approximations method for solving a class of nonlinear differential equations. We prove that these two methods are equivalent for solving these types of equations.     

On two numerical-analytic methods for the investigation of periodic solutions

We discuss two techniques useful in the investigation of periodic solutions of broad classes of non-linear non-autonomous ordinary differential equations, namely the trigonometric collocation and the method based upon periodic successive approximations. Supported in part by the Hungarian NFSR OTKA through Grant No. K68311.     

Solving the quasi-variational Signorini inequality by the method of successive approximations

The method of successive approximations is examined as a tool for solving the semicoercive quasi-variational Signorini inequality. The auxiliary problems with given friction arising at each step of this method are solved using the Uzawa method with an iterative proximal regularization of the modified Lagrangian functional.     

A successive approximations method of designing viscoplastic film structures

The solution of problems in which plasticity and creep have to be taken into account necessitates the formulation of cumber some nonlinear differential equations. Finding a solution (analytical or numerical) of these equations is a complex mathematical problem. In some cases, when more detailed data on the mechanical properties of the material in a complex stress state are available, the solution of such problems can be simplified by making use of the aging theory associated with the Tresca-St. Venant conditions of creep. A numerical solution is obtained in this case with the aid of geometrical conditions and equilibrium equations; the accuracy of the solution is determined by the number of approximations.Mekhanika Polimerov, Vol. 1, No. 3, pp. 137–144, 1965     

A method of successive approximations for solving the quasi-variational Signorini inequality

We solve the semicoercive quasi-variational Signorini inequality that corresponds to the contact problem with friction known in the elasticity theory by a method of successive approximations. For solving auxiliary problems with a given friction occurring on each outer step of the iterative process we use the Uzawa method based on iterative proximal regularization of a modified Lagrangian functional. We study the stabilization of the sequence of auxiliary finite-element solutions obtained on outer steps of the method of successive approximations and present results of numerical calculations.     

A successive approximations method for a cellular manufacturing problem

The problem of interest is to partition a collection of machines into production cells so that a given set of part-manufacturing requirements may be carried out optimally. In the present case transitions of parts between different cells is the only measure of machine partition goodness. The present formulation and approximate solution of this optimization problem is best described as one of successive approximations or as a one-at-a-time method. An initial cellular structure is taken and an easy part assignment optimization routine executed. With the part assignment fixed, a heuristic is employed to find an improved cell structure. These bipartite iterations continue until a convergence criterion is satisfied. Several small computer examples are provided and the straightforward requirements for large problem adaptation.     

Solutions to the random heat equation by the method of successive approximations

An iterative scheme for solving the random heat equation is proposed. Convergence of the method is established. Properties of the solution as well as error estimates are obtained. Indications as to possible application to nonlinear, inhomogeneous, time-dependent, random diffusion problems are given. A specific example of application to random diffusion in the unit interval is treated both analytically and numerically.     

Symbolic computing algorithm for the method of successive approximations for near-parabolic orbits

In this paper, an accurate algorithm for the method of successive approximations for near-parabolic orbits is established symbolically. Numerical applications are given for motion predictions at fifteen epochs between the years 66 to 1835 for Halley’s comet, and at fifteen epochs between the years 1417 to 1782 for Encke’s comet. Comparisons with the standard Gauss method [4] show that the present algorithm is very accurate and efficient for motion predications of near-parabolic orbits.     

Most of the successive approximations do converge

Let T be a locally compact Hausdorff space and E a Banach space. Let K(T, E) be the set of all continuous E-valued functions on T with compact support. We consider the representation of the second dual of K(T, E) when K(T, E) is normed with the usual sup norm. We demonstrate that an operator in the second dual of K(T, E) is, in a certain sense, approximable by an integral when computed over a certain subset of the dual of K(T, E).     

Revisions of constraint approximations in the successive QP method for nonlinear programming problems

In the last few years the successive quadratic programming methods proposed by Han and Powell have been widely recognized as excellent means for solving nonlinea programming problems. However, there remain some questions about their linear approximations to the constraints from both theoretical and empirical points of view. In this paper, we propose two revisions of the linear approximation to the constraints and show that the directions generated by the revisions are also descent directions of exact penalty functions of nonlinear programming problems. The new technique can cope better with bad starting points than the usual one.