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.     

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.     

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.     

On sufficient conditions for the convergence of successive approximations

We obtain sufficient conditions for the convergence of successive approximations to the solution of operator equations. We show how these conditions can be used in the case of nonlinear integral Hammerstein-type equations. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 854–859, June, 1999.     

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.     

On the convergence of successive approximations in dynamic programming with non-zero terminal reward

This paper considers the convergence of the finite-horizon optimal value functions of dynamic programming to the infinite-horizon optimal value function, when there is a non-zero terminal-reward function. The model and methods follow closely these used by Schäl in a recent paper, in which a terminal reward of zero was assumed. We first present convergence conditions that are direct extensions of Schäl's, then related conditions in which the terminal-reward function is an upper or lower bound for the infinite-horizon optimal value function. Some applications to problems in queueing control are mentioned briefly. We also comment on the relation between our conditions and the more restrictive conditions of strongly convergent and contractive models, and present a very general result concerning uniqueness of the solution to the infinite-horizon optimality equation.

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Zusammenfassung In der Arbeit wird die Konvergenz von Wertfunktionen dynamischer Optimierungsprobleme mit endlichem Planungshorizont gegen die Wertfunktionen bei unendlichem Planungshorizont betrachtet, wobei die Endauszahlung verschieden von Null ist. Modell und Vorgehensweise lehnen sich an entsprechende Resultate von Schäl an für den Fall, daß die Endauszahlung Null ist. Zunächst werden Konvergenzbedingungen angegeben, welche unmittelbare Erweiterungen der Schälschen Ergebnisse sind, gefolgt von Bedingungen, bei denen die Endauszahlung eine obere oder untere Schranke für die Endauszahlung bei unendlichem Planungshorizont ist. Einige Anwendungen auf Probleme der Steuerung von Warteschlangen werden erwähnt. Ferner werden der Zusammenhang zwischen unseren Bedingungen und den restriktiveren Bedingungen bei stark konvergenten und Kontraktions-Modellen erläutert und ein sehr allgemeines Modell über die Eindeutigkeit der Lösung der Optimalitätsgleichung bei unendlichem Planungshorizont angegeben.

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An earlier draft of this paper appeared as: On the Convergence of Successive Approximations and Uniqueness of the Solution to the Functional Equation of Dynamic Programming. IMSOR Report 2190, The Institute of Mathematical Statistics and Operations Research, The Technical University of Denmark, May 1977 (revised, July 1977).

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This research was partially supported by NATO Research Grant No. SRG.SS.5, administered by the NATO Special Programme Panel on Systems Science, and was begun while the author was guest professor at The Institute of Mathematical Statistics and Operations Research at The Technical University of Denmark, January to July, 1977. Further support was provided by the National Science Foundation under Grant No. ENG78-24420.     

Once again on the Samoilenko numerical-analytic method of successive periodic approximations

A new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear periodic systems of differential equations dx/dt = A(t) x+ ƒ(t, x) in the critical case is developed. The problem of the existence of solutions and their approximate construction is studied. Estimates for the convergence of successive periodic approximations are obtained. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 472–488, April, 2006.     

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

Mechanics of Composite Materials -     

On the convergence of the symmetric successive overrelaxation method

The symmetric successive overrelaxation (SSOR) iterative method is applied to the solution of the system of linear equations $\text{A}\text{x}\text{=}\text{b}$, where A is an n×n nonsingular matrix. We give bounds for the spectral radius of the SSOR iterative matrix when A is an Hermitian positive definite matrix, and when A is a nonsingular M-matrix. Then, we discuss the convergence of the SSOR iterative method associated with a new splitting of the matrix A which extends the results of Varga and Buoni [1].     

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.     

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.