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Generalizations of Shanks transformation and corresponding convergence acceleration algorithms via pfaffians

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Abstract

Firstly, a new sequence transformation that can be expressed in terms of a ratio of two pfaffians is derived based on a special kernel. It can be regarded as a direct generalization of Aitken’s Δ2 process from the point of view of pfaffians and then the corresponding convergence acceleration algorithm is constructed. Numerical examples with applications of this algorithm are also presented. Secondly, we find a way to generalize the Shanks transformation via pfaffians so that a larger class of new sequence transformations are derived. The corresponding recursive algorithms are also proposed.

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Acknowledgements

We thank Dr. S.H. Li for his useful conversations.

Funding

X.K. Chang was supported in part by the National Natural Science Foundation of China (Grant Nos. 11688101, 11731014, 11701550) and the Youth Innovation Promotion Association CAS. Y. He was supported in part by the National Natural Science Foundation of China (Grant No. 11971473), the Youth Innovation Promotion Association CAS, and the National Key Research and Development Program of China (No. 2020YFA0714200). X.B. Hu was supported in part by the National Natural Science Foundation of China (Grant Nos. 11931017, 11871336 and 12071447).

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Authors and Affiliations

  1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PO Box 2719, Beijing, 100190, People’s Republic of China

    Ya-Jie Liu, Xiang-Ke Chang & Xing-Biao Hu

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Ya-Jie Liu, Xiang-Ke Chang & Xing-Biao Hu

  3. Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan, 430071, People’s Republic of China

    Yi He

  4. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, People’s Republic of China

    Yi He

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  1. Ya-Jie Liu
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Correspondence to Xiang-Ke Chang.

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Liu, YJ., Chang, XK., He, Y. et al. Generalizations of Shanks transformation and corresponding convergence acceleration algorithms via pfaffians. Numer Algor 88, 1733–1756 (2021). https://doi.org/10.1007/s11075-021-01092-y

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