Numerical application of Euler's series transformation and its generalizations

Summary A nonlinear generalizationÊ _z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE _z and a twoparametric methodE . It is shown how to applyE orE _, to compute the valuef(z_o) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE _, are superior toÊ _z . A compact recursive algorithm is given for computingE andE _,.     

On two methods for accelerating convergence of series

Summary The Euler-Knopp transformation and a recently considered transformation, effective for entire function of order 1, are applied to series involving completely monotonic coefficients. Some properties of the resulting series are analyzed; these include uniform convergence with respect to the index, a priori and a posteriori estimates of the remainder. For the latter transformation a compact recursive algorithm is established which enables one to make effective use of the transformation. To illustrate the effectiveness of the transformations three applications, with examples, are included.     

Algorithmes pour suites non convergentes

Summary Two algorithms are proposed. For a given non-convergent sequence the first answers the question: How many cluster points does the sequence possess? The second one allows one to extract convergent sequences for any sequence with a finite number of cluster points.It was necessary to use the notion of strength of a cluster point. Two negative theorems show the necessity of the proposed hypotheses.

     

Some convergence results for asynchronous algorithms

Resumé Nous présentons dans cet article des résultats de convergence des algorithmes asynchrones basés essentiellement sur la notion classique de contraction.Nous généralisons, en particulier, tous les résultats de convergence de ces algorithmes qui font l'hypothèse de contraction en norme vectorielle qui récemment a été très souvant utilisée.Par ailleurs, l'hypothèse de contraction en norme vectorielle peut se trouver difficile, voire impossible à vérifier pour certains problèmes qui peuvent être cependant abordés dans le cadre de la contraction classique que nous adoptons.

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Some convergence results for asynchronous algorithms

Summary In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction.We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently.Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.

     

On the acceleration of limit periodic continued fractions

Summary A continued fraction (c.f.)K(a _n /1) is called limit periodic if . Fora anda(–,–1/4],a0, Thron-Waadeland (1980) examined a modification of a limit periodic c.f. for accelerating the convergence. This acceleration remains modest if thea _n converge only logarithmically. Thus it is proposed to apply an Euler summability method to the series equivalent to the c.f. Properties of the equivalent function are derived. These properties are used for choosing appropriate parameters for the summability method such that a considerable acceleration can be expected even if thea _n converge logarithmically.Dedicated to Prof. F.L. Bauer on the occasion of his 60th birthday     

Some supplementary results on the 1+\sqrt 2 order method for the solution of nonlinear equations

Summary Recently an iterative method for the solution of systems of nonlinear equations having at leastR-order 1+ for simple roots has been investigated by the author [7]; this method uses as many function evaluations per step as the classical Newton method. In the present note we deal with several properties of the method such as monotone convergence, asymptotic inclusion of the solution and convergence in the case of multiple roots.     

A method of solving nonlinear variational problems by nonlinear transformation of the objective functional. Part II

Summary The method of transformation of the objective functional is extended to solve nonlinear variational problems with non-differentiable objective functionals. The method is applied to the Bingham flow problem.     

A method of solving nonlinear variational problems by nonlinear transformation of the objective functional. Part I

Summary A general globally convergent iterative method for solving nonlinear variational problems is introduced. The method is applied to a temperature control problem and to the minimal surface problem. Several aspects of finite element implementation of the method are discussed.     

Repeated modifications of limitk-periodic continued fractions

Summary The advantages of using modified approximants for continued fractions, can be enhanced by repeating the modification process. IfK(a _n /b _n) is limitk-periodic, a natural choice for the modifying factors is ak-periodic sequence of right or wrong tails of the correspondingk-periodic continued fraction, if it exists. If the modified approximants thus obtained are ordinary approximants of a new limitk-periodic continued fraction, we repeat the process, if possible. Some examples where this process is applied to obtain a convergence acceleration are also given.     

On the numerical treatment of a small divisor problem

Summary Quasiperiodic solutions of perturbed integrable Hamiltonian equations such as weakly coupled harmonic oscillators can be found by constructing an appropriate coordinate transformation which leads to a small divisor problem. However the numerical difficulties are not merely caused by the small divisors but rather by the appearence of ghost solutions, which appear in any reasonable discretization of the problem. Our numerical treatment, based on a Newton-type iteration, guarantees an approximation of the relevant solution of the nonlinear problem. Numerical solutions are found up to a critical value of the coupling constant, which is much larger than the coupling constants allowed by the existence theory available so far.     

Kufarev's method for determining the Schwarz-Christoffel parameters

Summary We present an analysis of a method suggested by Kufarev for the numerical determination of the parameters involved in the Schwarz-Christoffel integral. We test the approach for certain polygonal domains including two with re-entrant corners and one with a cut. Finally, we consider the Motz problem from this point of view.     

Enlarging the domain of convergence for multiple shooting by the homotopy method

Summary The homotopy method is a frequently used technique in overcoming the local convergence nature of multiple shooting. In this paper sufficient conditions are given that guarantee the homotopy process to be feasible. The results are applicable to a class of two-point boundary value problems. Finally, the numerical solution of two practical problems arising in physiology is described.     

Improved difference schemes for the Dirichlet problem of Poisson's equation in the neighbourhood of corners

Summary The standard 5-point difference scheme for the model problem u=f on a special polygonal domain with given boundary values is modified at a few points in the neighbourhood of the corners in such a way that the order of convergence at interior points is the same as in the case of a smooth boundary. As a side result improved error bounds for the usual method in the neighbourhood of corners are given.     

A modified bisection algorithm for the determination of the eigenvalues of a symmetric tridiagonal matrix

Summary A modification to the well known bisection algorithm [1] when used to determine the eigenvalues of a real symmetric matrix is presented. In the new strategy the terms in the Sturm sequence are computed only as long as relevant information on the required eigenvalues is obtained. The resulting algorithm usingincomplete Sturm sequences can be shown to minimise the computational work required especially when only a few eigenvalues are required.The technique is also applicable to other computational methods which use the bisection process.     

Parameter tuning and repeated application of the IMT-type transformation in numerical quadrature

Summary The IMT rule, which is especially suited for the integration of functions with end-point singularities, is generalized by introducing parameters and also by repeatedly applying the parametrized IMT transformation. The quadrature formulas thus obtained are improved considerably both in efficiency and in robustness against end-point singularities. Asymptotic error estimates and numerical results are also given.     

On the convergence of the symmetric SOR method for matrices with red-black ordering

Summary We prove that if the matrixA has the structure which results from the so-called red-black ordering and ifA is anH-matrix then the symmetric SOR method (called the SSOR method) is convergent for 0<<2. In the special case thatA is even anM-matrix we show that the symmetric single-step method cannot be accelerated by the SSOR method. Symmetry of the matrixA is not assumed.     

On the sharpness of some upper bounds for the spectral radii of S.O.R. iteration matrices

Summary Sharpness is shown for three upper bounds for the spectral radii of point S.O.R. iteration matrices resulting from the splitting (i) of a nonsingularH-matrixA into the usualD–L–U, and (ii) of an hermitian positive definite matrixA intoD–L–U, whereD is hermitian positive definite andL=1/2(A–D+S) withS some skew-hermitian matrix. The first upper bound (which is related to the splitting in (i)) is due to Kahan [6], Apostolatos and Kulisch [1] and Kulisch [7], while the remaining upper bounds (which are related to the splitting in (ii)) are due to Varga [11]. The considerations regarding the first bound yield an answer to a question which, in essence, was recently posed by Professor Ridgway Scott: What is the largest interval in , 0, for which the point S.O.R. iterative method is convergent for all strictly diagonally dominant matrices of arbitrary order? The answer is, precisely, the interval (0, 1].Research supported in part by the Air Force Office of Scientific Research, and the Department of Energy     

The numerical solution of the inverse Stefan problem

Summary The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.     

Comparison of splittings used with the conjugate gradient algorithm

We consider several possible criteria for comparing splittings used with the conjugate gradient algorithm. We present sharp upper bounds for the error at each step of the algorithm and compare several widely used splittings with respect to their effect on these sharp upper bounds. We then consider a more stringent comparison test, and present necessary and sufficient conditions for the error at each step with one splitting to be smaller than that with another, for all pairs of corresponding initial guesses.