共查询到20条相似文献,搜索用时 334 毫秒
1.
Germund Dahlquist 《BIT Numerical Mathematics》1997,37(2):256-295
Three methods, old but not so well known, transform an infinite series into a complex integral over an infinite interval.
Gauss quadrature rules are designed for each of them. Various questions concerning their construction and application are
studied, theoretically or experimentally. They are so efficient that they should be considered for the development of software
for special functions.
Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation
of power series outside the circle of convergence, and to ill-conditioned power series. 相似文献
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M.A. Nyblom 《Journal of Number Theory》2004,105(1):49-59
As an application of Roth's theorem concerning the rational approximation of algebraic numbers, two sufficiency conditions are derived for an alternating series of rational terms to converge to a transcendental number. The first of these conditions represents an extension of an earlier condition of Sierpiński for the convergence of alternating series to irrational values. 相似文献
4.
对正弦和余弦富立叶级数,通过合并相邻同号项,使其重排成交错级数.讨论了重排形成的交错级数的敛散性.指出根据自变量x的不同取值,该交错级数可能是单调递减或周期递减的级数.按照莱布尼茨判定法提出了不同精度要求的级数项数的计算公式.选取一到三阶收敛的富立叶级数计算了不同比值精度及差值精度要求的级数项数.计算表明,在x的取值为2π的等分点时,富立叶级数的部分和随项数的增加单调地逼近其收敛值.在x的取值为其它点时,富立叶级数的部分和随项数的增加围绕收敛值上下变动,周期地逼近其收敛值.低收敛阶富立叶级数的收敛速度较慢.要达到0.01%的精度,一收敛阶富立叶级数需要数万项,二收敛阶富立叶级数也需要数百项.在不同计算点处,要达到相同的计算精度,需要的级数项数差别较大. 相似文献
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交替方向法是求解可分离结构变分不等式问题的经典方法之一, 它将一个大型的变分不等式问题分解成若干个小规模的变分不等式问题进行迭代求解. 但每步迭代过程中求解的子问题仍然摆脱不了求解变分不等式子问题的瓶颈. 从数值计算上来说, 求解一个变分不等式并不是一件容易的事情.因此, 本文提出一种新的交替方向法, 每步迭代只需要求解一个变分不等式子问题和一个强单调的非线性方程组子问题. 相对变分不等式问题而言, 我们更容易、且有更多的有效算法求解一个非线性方程组问题. 在与经典的交替方向法相同的假设条件下, 我们证明了新算法的全局收敛性. 进一步的数值试验也验证了新算法的有效性. 相似文献
6.
Truncated Fourier series and trigonometric interpolants converge slowly for functions with jumps in value or derivatives. The standard Fourier–Padé approximation, which is known to improve on the convergence of partial summation in the case of periodic, globally analytic functions, is here extended to functions with jumps. The resulting methods (given either expansion coefficients or function values) exhibit exponential convergence globally for piecewise analytic functions when the jump location(s) are known. Implementation requires just the solution of a linear system, as in standard Padé approximation. The new methods compare favorably in experiments with existing techniques. 相似文献
7.
R. E. Scraton 《BIT Numerical Mathematics》1989,29(2):356-360
The generalised Euler transformation is a powerful transformation of infinite series which can be used, in theory, for the acceleration of convergence and for analytic continuation. When the transformation is applied to a series with rounded coefficients, its behaviour can differ substantially from that predicted theoretically. In general, analytic continuation is impossible in this case. It is still possible, however, to use the transformation for acceleration of convergence, but some changes are necessary in the method of choosing the optimum parameter value. 相似文献
8.
Rudolf Wegmann 《Numerische Mathematik》1989,56(2-3):291-307
Summary The functional analytic principle of alternating projections is used to construct an iterative method for numerical conformal mapping of the unit disc onto regions with smooth boundaries. The result is a simple method which requires in each iterative step only two complex Fourier transforms. Local convergence can be proved using a theorem of Ostrowski. Convergence is linear. The asymptotic convergence factor is equal to the spectral radius of a certain operator. A version with overrelaxation as well as a discretized version are discussed along the same lines. For regions which are close to the unit disc convergence is fast. For some familiar regions the convergence factors can be calculated explicitly. Finally, the method is compared with Theodorsen's.Dedicated to the memory of Peter Henrici 相似文献
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Evgenia Kirillova 《International Journal of Mathematical Education in Science & Technology》2013,44(5):587-599
In this article we derive all salient properties of analytic functions, including the analytic version of the inverse function theorem, using only the most elementary convergence properties of series. Not even the notion of differentiability is required to do so. Instead, analytical arguments are replaced by combinatorial arguments exhibiting properties of formal power series. Along the way, we show how formal power series can be used to solve combinatorial problems and also derive some results in calculus with a minimum of analytical machinery. 相似文献
11.
Summary. The analytic dependence of Dirichlet-Neumann operators (DNO) with respect to variations of their domain of definition has
been successfully used to devise diverse computational strategies for their estimation. These strategies have historically
proven very competitive when dealing with small deviations from exactly solvable geometries, as in this case the perturbation
series of the DNO can be easily and recursively evaluated. In this paper we introduce a scheme for the enhancement of the
domain of applicability of these approaches that is based on techniques of analytic continuation. We show that, in fact, DNO
depend analytically on variations of arbitrary smooth domains. In particular, this implies that they generally remain analytic beyond the disk of convergence of their power
series representations about a canonical separable geometry. And this, in turn, guarantees that alternative summation mechanisms,
such as Padé approximation, can be effectively used to numerically access this extended domain of analyticity. Our method
of proof is motivated by our recent development of stable recursions for the coefficients of the perturbation series. Here, we again utilize this recursion as we compare and contrast
the performance of our new algorithms with that of previously advanced perturbative methods. The numerical results clearly
demonstrate the beneficial effect of incorporating analytic continuation procedures into boundary perturbation methods. Moreover,
the results also establish the superior accuracy and applicability of our new approach which, as we show, allows for precise
calculations corresponding to very large perturbations of a basic geometry.
Received October 10, 2000 / Revised version received January 21, 2002 / Published online June 17, 2002 相似文献
12.
《Journal of Computational and Applied Mathematics》2001,135(2):325-333
The alternating methods for solving the large system of linear equations Ax=b are investigated. The convergence and the monotone convergence theories for the alternating method are formulated when the coefficient matrix is an H-matrix or a monotone matrix. Sufficient conditions are established for the induced splitting by the alternating method to be a regular splitting. Furthermore, new comparison theorems which improve previous comparison theorems are proved and several concrete applications are given. 相似文献
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Heinz H. Bauschke J. Y. Bello Cruz Tran T. A. Nghia Hung M. Pha Xianfu Wang 《Numerical Algorithms》2016,73(1):33-76
We systematically study the optimal linear convergence rates for several relaxed alternating projection methods and the generalized Douglas-Rachford splitting methods for finding the projection on the intersection of two subspaces. Our analysis is based on a study on the linear convergence rates of the powers of matrices. We show that the optimal linear convergence rate of powers of matrices is attained if and only if all subdominant eigenvalues of the matrix are semisimple. For the convenience of computation, a nonlinear approach to the partially relaxed alternating projection method with at least the same optimal convergence rate is also provided. Numerical experiments validate our convergence analysis 相似文献
15.
Temple H Fay Gary L Walls 《International Journal of Mathematical Education in Science & Technology》2013,44(3):470-477
A result is discussed which permits the summing of series whose terms have more complicated sign patterns than simply alternating plus and minus. The Alternating Series Test, commonly taught in beginning calculus courses, is a corollary. This result, which is not difficult to prove, widens the series summable by beginning students and paves the way for understanding more advanced questions such as convergence of Fourier series. An elementary exposition is given of Dirichlet's Test for the convergence of a series and an elementary example suitable for a beginning calculus class and a more advanced example involving a Fourier series which is appropriate for an advanced calculus class are provided. Finally, two examples are discussed for which Dirichlet's Test does not apply and a general procedure is given for deciding the convergence or divergence of these and similar examples. 相似文献
16.
R. A. Lasuriya 《Mathematical Notes》2008,83(5-6):635-642
We obtain estimates of the rate of convergence of strong φ means of Λ methods of summation of Taylor series on some classes of analytic and bounded functions in the disk. 相似文献
17.
Conjugate-gradient acceleration provides a powerful tool for speeding up the convergence of a symmetrizable basic iterative method for solving a large system of linear algebraic equations with a sparse matrix. The object of this paper is to describe three generalizations of conjugate-gradient acceleration which are designed to speed up the convergence of basic iterative methods which are not necessarily symmetrizable. The application of the procedures to some commonly used basic iterative methods is described. 相似文献
18.
Shi-Jun Liao 《Communications in Nonlinear Science & Numerical Simulation》2009,14(5):2144-2159
An analytic method for strongly non-linear problems, namely the homotopy analysis method (HAM), is applied to give convergent series solution of non-similarity boundary-layer flows. As an example, the non-similarity boundary-layer flows over a stretching flat sheet are used to show the validity of this general analytic approach. Without any assumptions of small/large quantities, the corresponding non-linear partial differential equation with variable coefficients is transferred into an infinite number of linear ordinary differential equations with constant coefficients. More importantly, an auxiliary artificial parameter is used to ensure the convergence of the series solution. Different from previous analytic results, our series solutions are convergent and valid for all physical variables in the whole domain of flows. This work illustrates that, by means of the homotopy analysis method, the non-similarity boundary-layer flows can be solved in a similar way like similarity boundary-layer flows. Mathematically, this analytic approach is rather general in principle and can be applied to solve different types of non-linear partial differential equations with variable coefficients in science and engineering. 相似文献
19.
交替方向法适合于求解大规模问题.该文对于一类变分不等式提出了一种新的交替方向法.在每步迭代计算中,新方法提出了易于计算的子问题,该子问题由强单调的线性变分不等式和良态的非线性方程系统构成.基于子问题的精确求解,该文证明了算法的收敛性.进一步,又提出了一类非精确交替方向法,每步迭代计算只需非精确求解子问题.在一定的非精确条件下,算法的收敛性得以证明. 相似文献
20.
M. Turkyilmazoglu 《Applied Mathematical Modelling》2011,35(8):3879-3886
A novel approach is presented in this paper for approximate solution of parameterized unperturbed and singularly perturbed two-point boundary value problems. The problem is first separated into a simultaneous system regarding the unknown function and the parameter, and then a methodology based on the powerful homotopy analysis technique is proposed for the approximate analytic series solutions, whose convergence is guaranteed by optimally chosen convergence control parameters via square residual error. A convergence theorem is also provided. Several nonlinear problems are treated to validate the applicability, efficiency and accuracy of the method. Vicinity of the boundary layer is shown to be adequately treated and satisfactorily resolved by the method. Advantages of the method over the recently proposed conventional finite-difference or Runga–Kutta methods are also discussed. 相似文献