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1.
Summary The purpose of this paper is to construct a generalization of the Euler-Knopp transformation. Using this, one may recover previously known transformations, derive new transformations useful for numerical calculations and derive generating functions and other formulas of theoretical interest involving well-known functions.This work was supported by the Applied Mathematical Science Research Subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38  相似文献   

2.
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.  相似文献   

3.
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.
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4.
Summary A numerically applicable stepsize control for discrete continuation methods of orderp is derived on a theoretical basis. Both the theoretical results and the performance of the proposed algorithm are invariant under affine transformation of the nonlinear system to be solved. The efficiency and reliability of the method is demonstrated by solving three real life two-point boundary value problems using multiple shooting techniques. In two of the examples bifurcations occur and are significantly marked by sharp changes in the stepsize estimates.  相似文献   

5.
A general extrapolation algorithm   总被引:1,自引:0,他引:1  
Summary In this paper a general formalism for linear and rational extrapolation processes is developped. This formalism includes most of the sequence transformations actually used for convergence acceleration. A general recursive algorithm for implementing the method is given. Convergence results and convergence acceleration results are proved. The vector case and some other extensions are also studied.  相似文献   

6.
Summary An algorithm is presented for the computation of the second fundamental tensorV of a Riemannian submanifoldM ofR n . FromV the riemann curvature tensor ofM is easily obtained. Moreover,V has a close relation to the second derivative of certain functionals onM which, in turn, provides a powerful new tool for the computational determination of multiple bifurcation directions. Frequently, in applications, thed-dimensional manifoldM is defined implicitly as the zero set of a submersionF onR n . In this case, the principal cost of the algorithm for computingV(p) at a given pointpM involves only the decomposition of the JacobianDF(p) ofF atp and the projection ofd(d+1) neighboring points ontoM by means of a local iterative process usingDF(p). Several numerical examples are given which show the efficiency and dependability of the method.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThis work was in part supported by the National Science Foundation (DCR-8309926) and the Office of Naval Research (N-00014-80-C09455). The second author began some of the work while visiting the University of Heidelberg/Germany as an Alexander von Humboldt Senior U.S. Scientist  相似文献   

7.
Numerical computation of branch points in nonlinear equations   总被引:1,自引:0,他引:1  
Summary The numerical computation of branch points in systems of nonlinear equations is considered. A direct method is presented which requires the solution of one equation only. The branch points are indicated by suitable testfunctions. Numerical results of three examples are given.  相似文献   

8.
Summary This paper deals with the computation of branch points in ordinary differential equations. A direct numerical method is presented which requires the solution of only one boundary value problem. The method handles the general case of branching from a nontrivial solution which is a-prioriunknown. A testfunction is proposed which may indicate branching if used in continuation methods. Several real-life problems demonstrate the procedure.  相似文献   

9.
Summary For the numerical solution of two-point boundary value problems a shooting algorithm based on a Taylor series method is developed. Series coefficients are generated automatically by recurrence formulas. The performance of the algorithm is demonstrated by solving six problems arising in nonlinear shell theory, chemistry and superconductivity.  相似文献   

10.
Summary We shall consider a class of simple rational splines and their application to monotonic interpolation to monotonic data. Our method is situated between interpolation with the usual cubic splines and with monotone quadratic splines. A selection of numerical results is presented in Figs. 4–11.  相似文献   

11.
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.  相似文献   

12.
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  相似文献   

13.
Summary A possible way for parametrizing the solution path of the nonlinear systemH(u)=0, H: n+1 n consists of using the secant length as parameter. This idea leads to a quadratic constraint by which the parameter is introduced. A Newton-like method for computing the solution for a given parameter is proposed where the nonlinear system is linearized at each iterate, but the quadratic parametrizing equation is exactly satisfied. The localQ-quadratic convergence of the method is proved and some hints for implementing the algorithm are givenDedicated to Professor Lothar Collatz on the occasion of his 75th birthday  相似文献   

14.
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.  相似文献   

15.
Summary The classical proof that a separated Gershgorin disc contains an eigenvalue uses a continuity argument which is not applicable to Bauers generalized Gershgorin discs. In this paper the concept of positivity, generalized to complex vector spaces, is used to establish Gershgorins result and to improve the corresponding result for Bauers generalization.Dedicated to Professor F.L. Bauer on the occasion of his 60th birthday  相似文献   

16.
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.  相似文献   

17.
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.  相似文献   

18.
Summary We shall consider an application of simple exponential splines to the numerical solution of singular perturbation problem. The computational effort involved in our collocation method is less than that required for the other methods of exponential type.  相似文献   

19.
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.  相似文献   

20.
Summary In this paper we propose a numerical technique for the computation of Fourier transforms. It uses a bilateral expansion of the unknown transformed function with respect to Laguarre functions. The expansion coefficients are obtained via trigonometric interpolation and may be computed very efficiently by means of the Fast Fourier Transform. The convergence of the algorithm is analyzed and numerical results are presented which confirm that it works well.  相似文献   

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