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A new integrable convergence acceleration algorithm for computing Brezinski–Durbin–Redivo-Zaglia’s sequence transformation via pfaffians

Abstract

In the literature, most known sequence transformations can be written as a ratio of two determinants. But, it is not always this case. One exception is that the sequence transformation proposed by Brezinski, Durbin, and Redivo-Zaglia cannot be expressed as a ratio of two determinants. Motivated by this, we will introduce a new algebraic tool—pfaffians, instead of determinants in the paper. It turns out that Brezinski–Durbin–Redivo-Zaglia’s transformation can be expressed as a ratio of two pfaffians. To the best of our knowledge, this is the first time to introduce pfaffians in the expressions of sequence transformations. Furthermore, an extended transformation of high order is presented in terms of pfaffians and a new convergence acceleration algorithm for implementing the transformation is constructed. Then, the Lax pair of the recursive algorithm is obtained which implies that the algorithm is integrable. Numerical examples with applications of the algorithm are also presented.

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Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11331008, Grant No. 11371251, Grant no. 11201469, Grant No. 11571358).

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Correspondence to Shi-Hao Li.

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Chang, XK., He, Y., Hu, XB. et al. A new integrable convergence acceleration algorithm for computing Brezinski–Durbin–Redivo-Zaglia’s sequence transformation via pfaffians. Numer Algor 78, 87–106 (2018). https://doi.org/10.1007/s11075-017-0368-z

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  • DOI: https://doi.org/10.1007/s11075-017-0368-z

Keywords

  • Convergence acceleration algorithm
  • Sequence transformation
  • Pfaffian identities

Mathematics Subject Classification (2010)

  • Primary 65B05
  • 37K10