Convergence acceleration algorithms related to a generalized E-transformation and its particular cases

In this paper, a generalized E-transformation arising from the study of a generalization of sequence transformations and triangular recursion schemes is proposed. Three new algorithms, namely, the generalized E-algorithm, the generalized FS-algorithm and the generalized hungry type E-algorithm, are...

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Veröffentlicht in:	Japan journal of industrial and applied mathematics Jg. 30; H. 2; S. 263 - 285
Hauptverfasser:	He, Yi, Hu, Xing-Biao, Tam, Hon-Wah, Tsujimoto, Satoshi
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Japan Springer Japan 01.06.2013 Springer Nature B.V
Schlagworte:	Applications of Mathematics Computational Mathematics and Numerical Analysis Mathematics Mathematics and Statistics Original Paper rs-algorithm E-transformation 65B05 qd-algorithm G-algorithm 37K10 algorithm
ISSN:	0916-7005, 1868-937X
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Abstract	In this paper, a generalized E-transformation arising from the study of a generalization of sequence transformations and triangular recursion schemes is proposed. Three new algorithms, namely, the generalized E-algorithm, the generalized FS-algorithm and the generalized hungry type E-algorithm, are constructed for implementing the generalization of the E-transformation. Some convergence results of the generalized E-algorithm are obtained. In addition, some particular cases of the generalized E-transformation and the recursive algorithms for their computation are also studied.
AbstractList	In this paper, a generalized E-transformation arising from the study of a generalization of sequence transformations and triangular recursion schemes is proposed. Three new algorithms, namely, the generalized E-algorithm, the generalized FS-algorithm and the generalized hungry type E-algorithm, are constructed for implementing the generalization of the E-transformation. Some convergence results of the generalized E-algorithm are obtained. In addition, some particular cases of the generalized E-transformation and the recursive algorithms for their computation are also studied. In this paper, a generalized E-transformation arising from the study of a generalization of sequence transformations and triangular recursion schemes is proposed. Three new algorithms, namely, the generalized E-algorithm, the generalized FS-algorithm and the generalized hungry type E-algorithm, are constructed for implementing the generalization of the E-transformation. Some convergence results of the generalized E-algorithm are obtained. In addition, some particular cases of the generalized E-transformation and the recursive algorithms for their computation are also studied.[PUBLICATION ABSTRACT]
Author	Tsujimoto, Satoshi He, Yi Tam, Hon-Wah Hu, Xing-Biao
Author_xml	– sequence: 1 givenname: Yi surname: He fullname: He, Yi organization: Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences – sequence: 2 givenname: Xing-Biao surname: Hu fullname: Hu, Xing-Biao organization: LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences – sequence: 3 givenname: Hon-Wah surname: Tam fullname: Tam, Hon-Wah email: tam@comp.hkbu.edu.hk organization: Department of Computer Science, Hong Kong Baptist University – sequence: 4 givenname: Satoshi surname: Tsujimoto fullname: Tsujimoto, Satoshi organization: Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University
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References_xml	– reference: Brezinski, C.: A Bibliography on Continued Fractions, Padé Approximation. Sequence Transformation and Related Subjects. Prensas Universitarias de Zaragoza, Zaragoza (1991) – reference: SchneiderC.Vereinfachte Rekursionen zur Richardson-Extrapolation in SpezialfällenNumer. Math.1975241771843783420291.6500210.1007/BF01400966 – reference: ShanksD.Non linear transformations of divergent and slowly convergent sequencesJ. Math. Phys.195534142689010067.28602 – reference: Brezinski, C.: Convergence acceleration during the 20th century. J. Comput. Appl. Math. 122, 1–21: reprinted. In: Brezinski, C. (ed.) Numerical Analysis 2000, vol. 2, pp. 1–21. Interpolation and Extrapolation Elsevier, Amsterdam (2000) – reference: BrezinskiC.WalzG.Sequences of transformations and triangular recursion schemes, with applications in numerical analysisJ. Comut. Appl. Math.19913436138311025910732.6500410.1016/0377-0427(91)90095-2 – reference: Shanks, D.: An analogy between transient and mathematical sequences and some nonlinear sequence-to-sequence transforms suggested by it. Part I, Memorandum 9994, Naval Ordnance Laboratory, White Oak (1949) – reference: MeinardusG.TaylorG.D.Lower estimates for the error of the best uniform approximationJ. Approx. Theory.1976161501613997200317.4102110.1016/0021-9045(76)90044-7 – reference: FordW.F.SidiA.An algorithm for a generalization of the Richardson extrapolation processSIAM J. Numer. Anal.198724121212329090750627.6500110.1137/0724080 – reference: GrayH.L.AtchisonT.A.Nonlinear transformations related to the evaluation of improper integrals. ISIAM J. Numer. Anal.196743633712230960171.1440110.1137/0704032 – reference: Tsujimoto, S.: Molecule solution of hungry Volterra equations. In: Satsuma, J. (ed.) Soliton Theory and Its Applications. University of Tokyo, Japan, pp. 53–56 (1995). (In Japanese) – reference: Sidi, A.: Practical Extrapolation Methods. Cambridge University Press, Cambridge (2003) – reference: BrezinskiC.Extrapolation algorithms and Padé approximations: A historical surveyAppl. Numer. Math.19962029931814027040854.6500110.1016/0168-9274(95)00110-7 – reference: Brezinski, C.: Projection Methods for Systems of Equations. North-Holland, Amsterdam (1997) – reference: Brezinski, C., Redivo Zaglia, M.: Extrapolation Methods: Theory and Practice. North-Holland, Amsterdam (1991) – reference: Delahaye, J.P.: Sequence Transformations. Springer, Berlin (1988) – reference: GrayH.L.AtchisonT.A.McWilliamsG.V.Higher order G-transformationsSIAM J. Numer. Anal.197183653812889330236.6508910.1137/0708037 – reference: Wimp, J.: Sequence Transformations and Their Applications. Academic Press, New York (1981) – reference: Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (1992) – reference: WynnP.On a device for computing the em(Sn) transformationMTAC1956109196840560074.04601 – reference: BrezinskiC.A general extrapolation algorithmNumer. Math.1980351751875852450444.6500110.1007/BF01396314 – reference: HåvieT.Generalized Neville type extrapolation schemesBIT1979192042135377800404.6500110.1007/BF01930850 – reference: WenigerE.J.Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent seriesComput. Phys. Rep.19891018937110.1016/0167-7977(89)90011-7 – reference: Walz, G.: Asymptotics and Extrapolation. Akademie, Berlin (1996) – volume: 10 start-page: 189 year: 1989 ident: 102_CR7 publication-title: Comput. Phys. Rep. doi: 10.1016/0167-7977(89)90011-7 – volume: 4 start-page: 363 year: 1967 ident: 102_CR19 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0704032 – volume: 10 start-page: 91 year: 1956 ident: 102_CR23 publication-title: MTAC – volume: 34 start-page: 361 year: 1991 ident: 102_CR11 publication-title: J. Comut. Appl. Math. doi: 10.1016/0377-0427(91)90095-2 – volume: 8 start-page: 365 year: 1971 ident: 102_CR20 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0708037 – volume: 24 start-page: 177 year: 1975 ident: 102_CR15 publication-title: Numer. Math. doi: 10.1007/BF01400966 – ident: 102_CR4 doi: 10.1017/CBO9780511546815 – volume: 34 start-page: 1 year: 1955 ident: 102_CR22 publication-title: J. Math. Phys. doi: 10.1002/sapm19553411 – volume: 16 start-page: 150 year: 1976 ident: 102_CR14 publication-title: J. Approx. Theory. doi: 10.1016/0021-9045(76)90044-7 – ident: 102_CR17 – volume: 20 start-page: 299 year: 1996 ident: 102_CR9 publication-title: Appl. Numer. Math. doi: 10.1016/0168-9274(95)00110-7 – ident: 102_CR3 doi: 10.1007/978-3-642-61347-0 – ident: 102_CR18 – ident: 102_CR10 doi: 10.1016/S0377-0427(00)00360-5 – volume: 24 start-page: 1212 year: 1987 ident: 102_CR16 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0724080 – ident: 102_CR21 – ident: 102_CR1 – ident: 102_CR2 – ident: 102_CR8 doi: 10.1007/978-3-642-58169-4 – volume: 19 start-page: 204 year: 1979 ident: 102_CR13 publication-title: BIT doi: 10.1007/BF01930850 – ident: 102_CR5 – ident: 102_CR6 – volume: 35 start-page: 175 year: 1980 ident: 102_CR12 publication-title: Numer. Math. doi: 10.1007/BF01396314
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Title	Convergence acceleration algorithms related to a generalized E-transformation and its particular cases
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