An Extended Multistep Shanks Transformation and Convergence Acceleration Algorithm with Their Convergence and Stability
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Numer. Math. DOI 10.1007/s00211-013-0549-1
An Extended Multistep Shanks Transformation and Convergence Acceleration Algorithm with Their Convergence and Stability Analysis Jian-Qing Sun · Xiang-Ke Chang · Yi He · Xing-Biao Hu
Received: 8 July 2011 / Revised: 1 February 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract The molecule solution of an extended discrete Lotka–Volterra equation is constructed, from which a new sequence transformation is proposed. A convergence acceleration algorithm for implementing this sequence transformation is found. It is shown that our new sequence transformation accelerates some kinds of linearly convergent sequences and factorially convergent sequences with good numerical stability. Some numerical examples are also presented. Mathematics Subject Classification
65B05 · 37K10 · 39A14
1 Introduction Some intimate relations between certain numerical algorithms and integrable systems have been revealed in recent years, which leads to a reinvestigation of both objects.
J.-Q. Sun (B) School of Mathematical Sciences, Ocean University of China, Qingdao 266100, People’s Republic of China e-mail: [email protected] X.-K. Chang · X.-B. Hu LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences, P.O.Box 2719, Beijing 100190, People’s Republic of China X.-K. Chang University of Chinese Academy of Sciences, Beijing, People’s Republic of China Y. He Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China
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J.-Q. Sun et al.
On the one hand, many algorithms in numerical analysis when considered as dynamical systems, have a variety of interesting dynamical behavior (see [17]). One of the intriguing properties is integrability, which distinguishes these numerical schemes from others, and attracts workers in the field of integrable systems to study integrable properties of numerical algorithms. In the literature, integrability of numerical algorithms always appears in various guises such as properties of invariance, compatibility and identity. To be precise, the notion of invariance refers to the existence of a sufficient number of conserved quantities, which is a key feature of the Liouville integrability [4] in some sense. And the compatibility means that the equation can be seen as the compatibility condition of some linear problems. Finally, the identity property indicates that, essentially, integrable equations are some kind of determinantal (or Pfaffian) identities on so-called τ -function level [23,39]. For example, Gauss arithmetic-geometric mean algorithm [3,6,31] for computing the first elliptic integral can be viewed as an illustration of invariance. In fact, this algorithm corresponds to a discrete-time integrable equation, with the corresponding elliptic integral as its conserved quantity. In addition, the qd-algorithm [21,38], which plays a significant role in the theory of formal orthogonal polynomials and Padé approximants [5,9,35], is nothing but the
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