The Toda molecule equation and the 𝜖-algorithm

Nagai, Atsushi; Tokihiro, Tetsuji; Satsuma, Junkichi

doi:10.1090/S0025-5718-98-00987-9

The Toda molecule equation and the $\varepsilon$ -algorithm
HTML articles powered by AMS MathViewer

by Atsushi Nagai, Tetsuji Tokihiro and Junkichi Satsuma PDF

Math. Comp. 67 (1998), 1565-1575 Request permission

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.

References

Masaharu Arai, Kazuo Okamoto, and Yoshinori Kametaka, Aitken-Steffensen acceleration and a new addition formula for Fibonacci numbers, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 1, 5–7. MR 839793
George A. Baker Jr. and Peter Graves-Morris, Padé approximants. Part I, Encyclopedia of Mathematics and its Applications, vol. 13, Addison-Wesley Publishing Co., Reading, Mass., 1981. Basic theory; With a foreword by Peter A. Carruthers. MR 635619
Friedrich L. Bauer, The quotient-difference and epsilon algorithms, University of Wisconsin Press, Madison, Wis., 1959. Edited by R. E. Langer. Publication no. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin. MR 0102594
Claude Brezinski, Padé-type approximation and general orthogonal polynomials, International Series of Numerical Mathematics, vol. 50, Birkhäuser Verlag, Basel-Boston, Mass., 1980. MR 561106, DOI 10.1007/978-3-0348-6558-6
Claude Brezinski and Michela Redivo Zaglia, Extrapolation methods, Studies in Computational Mathematics, vol. 2, North-Holland Publishing Co., Amsterdam, 1991. Theory and practice; With 1 IBM-PC floppy disk (5.25 inch). MR 1140920
Annie Cuyt, The mechanism of the multivariate Padé process, Padé approximation and its applications, Bad Honnef 1983 (Bad Honnef, 1983) Lecture Notes in Math., vol. 1071, Springer, Berlin, 1984, pp. 95–103. MR 757550, DOI 10.1007/BFb0099611
H. Flaschka, On the Toda lattice. II. Inverse-scattering solution, Progr. Theoret. Phys. 51 (1974), 703–716. MR 408648, DOI 10.1143/PTP.51.703
W. B. Gragg, The Padé table and its relation to certain algorithms of numerical analysis, SIAM Rev. 14 (1972), 1–16. MR 305563, DOI 10.1137/1014001
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.
R. Hirota, Direct Method in Soliton Theory, Iwanami, Tokyo, 1992 [in Japanese].
Ryogo Hirota, Satoshi Tsujimoto, and Tatsuya Imai, Difference scheme of soliton equations, Future directions of nonlinear dynamics in physical and biological systems (Lyngby, 1992) NATO Adv. Sci. Inst. Ser. B: Phys., vol. 312, Plenum, New York, 1993, pp. 7–15. MR 1262147
Yoshimasa Nakamura, A tau-function of the finite nonperiodic Toda lattice, Phys. Lett. A 195 (1994), no. 5-6, 346–350. MR 1304653, DOI 10.1016/0375-9601(94)90040-X
Yoshimasa Nakamura, The BCH-Goppa decoding as a moment problem and a tau function over finite fields, Phys. Lett. A 223 (1996), no. 1-2, 75–81. MR 1421513, DOI 10.1016/S0375-9601(96)00718-9
A. Nagai and J. Satsuma, Discrete soliton equations and convergence acceleration algorithms, Phys. Lett. A 209 (1995), no. 5-6, 305–312. MR 1368590, DOI 10.1016/0375-9601(95)00865-9
V. Papageorgiou, B. Grammaticos, and A. Ramani, Integrable lattices and convergence acceleration algorithms, Phys. Lett. A 179 (1993), no. 2, 111–115. MR 1230809, DOI 10.1016/0375-9601(93)90658-M
O. Perron, Die Lehre von den Kettenbrüchen, Taubner, Leipzig, 1929.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Kiyoshi Sogo, Toda molecule equation and quotient-difference method, J. Phys. Soc. Japan 62 (1993), no. 4, 1081–1084. MR 1221214, DOI 10.1143/JPSJ.62.1081
W. W. Symes, The $QR$ algorithm and scattering for the finite nonperiodic Toda lattice, Phys. D 4 (1981/82), no. 2, 275–280. MR 653781, DOI 10.1016/0167-2789(82)90069-0
M. Toda, Waves in Nonlinear Lattice, Prog. Thoer. Phys. Suppl. 45 (1970) 174–200.
H. Togawa, Matrix Computations, Ohm, Tokyo, 1971 [in Japanese].
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
D. S. Watkins and L. Elsner, On Rutishauser’s approach to self-similar flows, SIAM J. Matrix Anal. Appl. 11 (1990), no. 2, 301–311. MR 1041249, DOI 10.1137/0611020
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
P. Wynn, Confluent forms of certain non-linear algorithms, Arch. Math. 11 (1960), 223–236. MR 128068, DOI 10.1007/BF01236936
P. Wynn, A note on a confluent form of the $\varepsilon$-algorithm, Arch. Math. 11 (1960), 237–240. MR 128069, DOI 10.1007/BF01236937
P. Wynn, Upon a second confluent form of the $\varepsilon$-algorithm, Proc. Glasgow Math. Assoc. 5 (1962), 160–165 (1962). MR 139253, DOI 10.1017/S2040618500034535
P. Wynn, On a connection between the first and second confluent forms of the $\varepsilon$-algorithm, Nieuw Arch. Wisk. (3) 11 (1963), 19–21. MR 149147
Peter Wynn, Partial differential equations associated with certain non-linear algorithms, Z. Angew. Math. Phys. 15 (1964), 273–289 (English, with German summary). MR 166944, DOI 10.1007/BF01607018