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The Toda molecule equation and the $\varepsilon$-algorithm


Authors: Atsushi Nagai, Tetsuji Tokihiro and Junkichi Satsuma
Journal: Math. Comp. 67 (1998), 1565-1575
MSC (1991): Primary 58F07, 65B10
DOI: https://doi.org/10.1090/S0025-5718-98-00987-9
MathSciNet review: 1484902
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Abstract: One of the well-known convergence acceleration methods, the $\varepsilon$-algorithm is investigated from the viewpoint of the Toda molecule equation. It is shown that the error caused by the algorithm is evaluated by means of solutions for the equation. The acceleration algorithm based on the discrete Toda molecule equation is also presented.


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  • 1. M. Arai, K. Okamoto and Y. Kametaka, Aitken-Steffenson Acceleration and a New Addition Formula for Fibonacci Numbers, Proc. Japan. Acad. 62 Ser. A (1986) 5-7. MR 87g:11028
  • 2. G. A. Baker Jr. and P. Graves-Morris, Padé Approximants Part I : Basic Theory, Addison-Wesley, Massachusetts, 1981. MR 83a:41009a
  • 3. F. L. Bauer, The quotient-difference and epsilon algorithms, On Numerical Approximation, R. E. Langer, ed., University of Wisconsin Press, Madison, 1959, 361-370. MR 21:1384
  • 4. C. Brezinski, Padé-Type Approximation and General Orthogonal Polynomials, Birkhäuser-Verlag, Basel, 1980. MR 82a:41017
  • 5. C. Brezinski and M. Redivo Zaglia, Extrapolation Methods. Theory and Practice, North-Holland, Amsterdam, 1991. MR 93d:65001
  • 6. A. Cuyt, The Mechanism of the Multivariate Padé Process, Lecture notes in mathematics Vol. 1071, Padé Approximation and its Applications, H. Werner and H. J. Bünger, ed., Springer, Berlin, 1984, 95-103. MR 85j:41034
  • 7. H. Flaschka, The Toda lattice. II. Existence of Integrals, Phys. Rev. B, 9 (1974) 1924-1925. MR 53:12412
  • 8. W. B. Gragg, The Padé table and its relation to certain algorithms of numerical analysis, SIAM Review 14 (1972) 1-62. MR 46:4693
  • 9. R. Hirota and J. Satsuma, A Variety of Nonlinear Network Equations Generated from the Bäcklund Transformation for the Toda Lattice, Suppl. Prog. Theor. Phys. 59 (1976) 64-100.
  • 10. R. Hirota, Direct Method in Soliton Theory, Iwanami, Tokyo, 1992 [in Japanese].
  • 11. R. Hirota, S. Tsujimoto and T. Imai, Difference scheme of soliton equations, RIMS Kokyuroku, 822 (1992) 144-152. MR 95j:58070
  • 12. Y. Nakamura, A tau-function of the finite nonperiodic Toda lattice, Phys. Lett. A 195 (1994) 346-350. MR 95j:58078
  • 13. Y. Nakamura, The BCH-Goppa decoding as a moment problem and a tau-function over finite fields, Phys. Lett. A 223 (1996) 75-81. MR 97k:94074
  • 14. A. Nagai and J. Satsuma, Discrete soliton equations and convergence acceleration algorithms, Phys. Lett. A 209 (1995) 305-312. MR 96h:65006
  • 15. V. Papageorgiou, B. Grammaticos and A. Ramani, Integrable lattices and convergence acceleration algorithms Phys. Lett. A 179 (1993) 111-115. MR 94b:65009
  • 16. O. Perron, Die Lehre von den Kettenbrüchen, Taubner, Leipzig, 1929.
  • 17. H. Rutishauser, Der Quotienten-Differenzen-Algorithmus, Z.A.M.P. 5 (1954) 233-251. MR 16:176c
  • 18. D. Shanks, Nonlinear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955) 1-42. MR 16:961e
  • 19. K. Sogo, Toda Molecule Equation and Quotient-Difference Method, J. Phys. Soc. Jpn. 62 (1993) 1081-1084. MR 94i:82022
  • 20. W. W. Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice, Physica 4D (1982) 275-280. MR 83h:58053
  • 21. M. Toda, Waves in Nonlinear Lattice, Prog. Thoer. Phys. Suppl. 45 (1970) 174-200.
  • 22. H. Togawa, Matrix Computations, Ohm, Tokyo, 1971 [in Japanese].
  • 23. H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, Bronx, N. Y. 1948. MR 10:32d
  • 24. D. S. Watkins and L. Elsner, On Rutishauser's Approach to Self-Similar Flows, SIAM J. Matrix Anal. Appl. 11 (1990) 301-311. MR 91e:58145
  • 25. P. Wynn, On a device for computing the $e_m(S_n)$ transformations, Math. Tables Aids Comput. 10 (1956) 91-96. MR 18:801e
  • 26. P. Wynn, Confluent Forms of Certain Non-linear Algorithms, Arch. Math. 11 (1960) 223-236. MR 23:B1113
  • 27. P. Wynn, A Note on a Confluent Form of the $\varepsilon $-algorithm, Arch. Math. 11 (1960) 237-240. MR 23:B1114
  • 28. P. Wynn, Upon a Second Confluent Form of the $\varepsilon $-algorithm, Proc. Glasgow Math. Soc. 5 (1962) 160-165. MR 25:2689
  • 29. P. Wynn, On a Connection between the First and Second Confluent Forms of the $\varepsilon $-algorithm, Nieuw Archief voor Wiskunde 11 (1963) 19-21. MR 26:6641
  • 30. P. Wynn, Partial Differential Equations Associated with Certain Non-Linear Algorithms, Z. Angew. Math. Phys. 15(1964) 273-289. MR 29:4217

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Additional Information

Atsushi Nagai
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153, Japan
Email: slime@poisson.ms.u-tokyo.ac.jp

Tetsuji Tokihiro
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153, Japan
Email: toki@sunflower.t.u-tokyo.ac.jp

Junkichi Satsuma
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153, Japan
Email: satsuma@poisson.ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0025-5718-98-00987-9
Keywords: Toda molecule equation, $\varepsilon$-algorithm, Pad\'{e} approximation, continued fraction
Received by editor(s): May 20, 1996
Received by editor(s) in revised form: November 5, 1996, and February 13, 1997
Article copyright: © Copyright 1998 American Mathematical Society

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