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1.
We give a formula for efficient steplength control in numerical integration, based on comparison of the numerical solution with a solution of lower order.  相似文献   

2.
We derive a general two-point integral quadrature formula using the concept of harmonic polynomials. An improved version of Guessab and Schmeisser’s result is given with new integral inequalities involving functions whose derivatives belong to various classes of functions (LpLp spaces, convex, concave, bounded functions). Furthermore, several special cases of polynomials are considered, and the generalization of well-known two-point quadrature formulae, such as trapezoid, perturbed trapezoid, two-point Newton–Cotes formula, two-point Maclaurin formula, midpoint, are obtained.  相似文献   

3.
Recently Dolezal and Tewarson [2], and Papamichael and Soares [3] have considered the cubic spline-on-spline for the purpose of the approximating the derivatives y(2),y(3), and y(4). In this paper their ideas have been extended and the quadratic spline-on-spline has been established for the same purpose. This technique yields better results than the traditional process using a single quadratic spline.  相似文献   

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Zusammenfassung Es wird ein Schrittverfahren zur numerischen Integration von gestörten linearen Differentialgleichungssystemen aufgestellt. Die Grundlage dazu bildet die Definition einer Folge von ganzen Funktionen, die keine Polynome sind, aber eine einfache Potenzreihendarstellung besitzen. Im Gegensatz zu den üblichen Potenzreihenverfahren, wird nicht nach Potenzen der unabhängigen Variabeln, sondern nach diesen Funktionen entwickelt. Das Integrationsverfahren hat die Eigenschaft, dass die ungestörten Differentialgleichungen ohne Diskretisationsfehler integriert werden, und dass die Eigenwerte der Koeffizientenmatrix nicht berechnet werden müssen. Die Verbesserung gegenüber der gewöhnlichen Potenzreihenmethode wird durch asymptotische Formeln für die Residuen und durch numerische Beispiele belegt.Nach einer Einleitung über Ziel und Zweck der Arbeit wird die Methode anhand eine: gestörten linearen Differentialgleichung zweiter Ordnung illustriert. Anschliessend wird das Integrationsverfahren auf beliebige gestörte lineare Systeme verallgemeinert.  相似文献   

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On the method for numerical integration of Clenshaw and Curtis   总被引:1,自引:0,他引:1  
Letf(x) be a function, defined and well behaved on the finite intervalaxb Clenshaw andCurtis [1] have given a method for the numerical integration off(x) froma tob, based on the approximation off(x) with a finite series of Chebyshev polynomials. We show that this method is asymptotically equivalent to using the trapezoïdal rule for integratingg(y)=f(cosy).  相似文献   

9.
Algorithms to calculate the weights of quadrature formulas for regular, weakly singular, and singular integrals are constructed and analyzed. The algorithms are easily realized by computer for any number, multiplicity, and distribution of quadrature nodes, and for systems of power, trigonometric, and hyperbolic basis functions as well as a given strictly monotone function used as the integration variable.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 3–14, 1989.  相似文献   

10.
Summary Gaussian quadrature is inappropriate for evaluating a definite integral with logarithmic end-point singularities that occurs in the preceding paper [1]. Some modern ways to evaluate the integral are indicated.
Zusammenfassung Die Gauss'sche Quadraturformel ist nicht das geeignete Mittel zur Auswertung eines in der vorstehenden Arbeit [1] vorkommenden Integrals, dessen Integrand in den Endpunkten logarithmische Singularitäten hat. Wir berechnen das Integral mit Hilfe von drei zeitgemäßen Quadratur-Algorithmen.
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11.
The paper develops a construction for finding fully symmetric integration formulas of arbitrary degree 2k+1 inn-space such that the number of evaluation points isO((2n)k)/k!),n . Formulas of degrees 3, 5, 7, 9, are relatively simple and are presented in detail. The method has been tested by obtaining some special formulas of degrees 7, 9 and 11 but these are not presented here.  相似文献   

12.
An error formula for numerical differentiation   总被引:1,自引:0,他引:1  
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A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.  相似文献   

16.
It is asserted on the basis of empirical evidence supported in some cases by theoretical analysis, that in the numerical integration of an integrand which is singular in the function-analytic sense at a point at which the function is defined, it is preferable to use an integration rule which does not include the singular point among its abscissas.  相似文献   

17.
In this study, approximating the finite Hilbert transform are given for absolutely continuous mappings. Then, some numerical experiments for the obtained approximation are also presented.  相似文献   

18.
Since the advantages of quasi-Monte Carlo methods vanish when the dimension of the basic space increases, the question arises whether there are better methods than the classical Monte Carlo in large or infinite-dimensional basic spaces. We study here the use of the shift operator with the pointwise ergodic theorem whose implementation is particularly interesting. After recalling the theoretical results on the speed of convergence in a form useful for applications, we give sufficient criteria for the law of iterated logarithm in several cases and, in particular, in situations involving the Wiener space.  相似文献   

19.
Extrapolation methods have been used for many years for numerical integration. The most well-known of these methods is Romberg integration. A survey by Joyce on the use of extrapolation in numerical analysis appeared in 1971 in which a substantial portion is devoted to numerical integration. In this paper, we shall survey progress made in this field since 1971. The topics surveyed include partition-extrapolation methods for dealing with singular integrands, the work of Lyness and others in generating asymptotic expansions for the error functional in one and several dimensions, the work of de Doncker and others on adaptive extrapolation and the work of Sidi and others on the evaluation of highly oscillatory infinite integrals by extrapolation. Other extrapolation techniques will be mentioned briefly.  相似文献   

20.
Avram Sidi 《PAMM》2007,7(1):2020019-2020020
We discuss some recent developments in the use of class Sm variable transformations for improving the accuracy of the trapezoidal rule approximations in numerical integration via the trapezoidal rule. These concern optimal choices of m. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

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