Convergence acceleration during the 20th century

Journal of Computational and Applied Mathematics

Volumn 122, Issue 1, 2000, Pages 1-21

  Brezinski, C ^a  

^a UNIV LILLE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; EXTRAPOLATION; INTERPOLATION; ITERATIVE METHODS; LINEAR EQUATIONS; MATHEMATICAL TRANSFORMATIONS; SET THEORY; VECTORS;

SEQUENCE TRANSFORMATIONS;

CONVERGENCE OF NUMERICAL METHODS;

MATHEMATICAL METHOD;

EID: 0034289685     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00360-5     Document Type: Article

References (156)

1. A.C. Aitken, On Bernoulli's numerical solution of algebraic equations, Proc. Roy. Soc. Edinburgh 46 (1926) 289-305.
2. H. Andoyer, Interpolation, in: J. Molk (Ed.), Encyclopédie des Sciences Mathématiques Pures et Appliquées, Tome I, Vol. 4, Fasc. 1, I-21, Gauthier-Villars, Paris, 1904-1912, pp.127-160; (reprint by Editions Gabay, Paris, 1993).
3. F.L. Bauer, La méthode d'intégration numérique de Romberg, in: Colloque sur l'Analyse Numérique, Librairie Universitaire, Louvain, 1961, pp. 119-129.
4. B. Beckermann, A connection between the E-algorithm and the epsilon-algorithm, in: C. Brezinski (Ed.), Numerical and Applied Mathematics, Baltzer, Basel, 1989, pp. 443-446.
5. 2 d'Aitken et Comparaison de Procédés d'Accélération de la Convergence, Thèse 3ème cycle, Université de Lille I, 1987.
6. M.D. Benchiboun, Extension of Henrici's method to matrix sequences, J. Comput. Appl. Math. 75 (1996) 1-21.
7. A. Benazzouz, Quasilinear sequence transformations, Numer. Algorithms 15 (1997) 275-285.
8. A. Benazzouz, GL(E)-quasilinear transformations and acceleration, Appl. Numer. Math. 27 (1998) 109-122.
9. A. Berlinet, Sequence transformations as statistical tools, Appl. Numer. Math. 1 (1985) 531-544.
10. A.H. Bentbib, Acceleration of convergence of interval sequences, J. Comput. Appl. Math. 51 (1994) 395-409.
11. N. Bogolyubov, N. Krylov, On Rayleigh's principle in the theory of differential equations of mathematical physics and upon Euler's method in the calculus of variation, Acad. Sci. Ukraine (Phys. Math.) 3 (1926) 3-22 (in Russian).
12. H.C. Bolton, H.I. Scoins, Eigenvalues of differential equations by finite-difference methods, Proc. Cambridge Philos. Soc. 52 (1956) 215-229.
13. C. Brezinski, Application de l'ε-algorithme à la résolution des systèmes non linéaires, C.R. Acad. Sci. Paris 271 A (1970) 1174-1177.
14. C. Brezinski, Accélération de suites à convergence logarithmique, C. R. Acad. Sci. Paris 273 A (1971) 727-730.
15. C. Brezinski, Etude sur les ε et ρ-algorithmes, Numer. Math. 17 (1971) 153-162.
16. C. Brezinski, L'ε-algorithme et les suites totalement monotones et oscillantes, C.R. Acad. Sci. Paris 276 A (1973) 305-308.
17. C. Brezinski, Généralisation de la transformation de Shanks, de la table de Padé et de l'ε-algorithme, Calcolo 12 (1975) 317-360.
18. C. Brezinski, Padé-Type Approximation and General Orthogonal Polynomials, Birkhäuser, Basel, 1980.
19. C. Brezinski, A general extrapolation algorithm, Numer. Math. 35 (1980) 175-187.
20. C. Brezinski, Recursive interpolation, extrapolation and projection, J. Comput. Appl. Math. 9 (1983) 369-376.
21. C. Brezinski, Error control in convergence acceleration processes, IMA J. Numer. Anal. 3 (1983) 65-80.
22. C. Brezinski, Prediction properties of some extrapolation methods, Appl. Numer. Math. 1 (1985) 457-462.
23. C. Brezinski, Composite sequence transformations, Numer. Math. 46 (1985) 311-321.
24. C. Brezinski, A. Lembarki, Acceleration of extended Fibonacci sequences, Appl. Numer. Math. 2 (1986) 1-8.
25. C. Brezinski, A new approach to convergence acceleration methods, in: A. Cuyt (Ed.), Nonlinear Numerical Methods and Rational Approximation, Reidel, Dordrecht, 1988, pp. 373-405.
26. C. Brezinski, Quasi-linear extrapolation processes, in: R.P. Agarwal et al. (Eds.), Numerical Mathematics, Singapore 1988, International Series of Numerical Mathematics, Vol. 86, Birkhäuser, Basel, 1988, pp. 61-78.
27. C. Brezinski, A survey of iterative extrapolation by the E-algorithm, Det Kong. Norske Vid. Selsk. Skr. 2 (1989) 1-26.
28. C. Brezinski, A Bibliography on Continued Fractions, Padé Approximation, Extrapolation and Related Subjects, Prensas Universitarias de Zaragoza, Zaragoza, 1991.
29. C. Brezinski, History of Continued Fractions and Padé Approximants, Springer, Berlin, 1991.
30. C. Brezinski, The generalizations of Newton's interpolation formula due to Mühlbach and Andoyer, Electron Trans. Numer. Anal. 2 (1994) 130-137.
31. C. Brezinski, Extrapolation algorithms and Padé approximations: a historical survey, Appl. Numer. Math. 20 (1996) 299-318.
32. C. Brezinski, Variations on Richardson's method and acceleration, in: Numerical Analysis, A Numerical Analysis Conference in Honour of Jean Meinguet, Bull. Soc. Math. Belgium 1996, pp. 33-44.
33. C. Brezinski, Projection Methods for Systems of Equations, North-Holland, Amsterdam, 1997.
34. C. Brezinski, Vector sequence transformations: methodology and applications to linear systems, J. Comput. Appl. Math. 98 (1998) 149-175.
35. C. Brezinski, Error estimates for the solution of linear systems, SIAM J. Sci. Comput. 21 (1999) 764-781.
36. C. Brezinski, Acceleration procedures for matrix iterative methods, Numer. Algorithms, to appear.
37. C. Brezinski, J.P. Delahaye, B. Germain-Bonne, Convergence acceleration by extraction of linear subsequences, SIAM J. Numer. Anal. 20 (1983) 1099-1105.
38. C. Brezinski, A.C. Matos, A derivation of extrapolation algorithms based on error estimates, J. Comput. Appl. Math. 66 (1996) 5-26.
39. C. Brezinski, M. Redivo Zaglia, Construction of extrapolation processes, Appl. Numer. Math. 8 (1991) 11-23.
40. C. Brezinski, M. Redivo Zaglia, Extrapolation Methods, Theory and Practice, North-Holland, Amsterdam, 1991.
41. C. Brezinski, M. Redivo Zaglia, A general extrapolation procedure revisited, Adv. Comput. Math. 2 (1994) 461-477.
42. C. Brezinski, M. Redivo Zaglia, Vector and matrix sequence transformations based on biorthogonality, Appl. Numer. Math. 21 (1996) 353-373.
43. C. Brezinski, M. Redivo Zaglia, G. Rodriguez, S. Seatzu, Extrapolation techniques for ill-conditioned linear systems, Numer. Math. 81 (1998) 1-29.
44. C. Brezinski, A.C. Rieu, The solution of systems of equations using the vector ε-algorithm, and an application to boundary value problems, Math. Comp. 28 (1974) 731-741.
45. C. Brezinski, A. Salam, Matrix and vector sequence transformation revisited, Proc. Edinburgh Math. Soc. 38 (1995) 495-510.
46. C. Brezinski, G. Walz, Sequences of transformations and triangular recursion schemes with applications in numerical analysis, J. Comput. Appl. Math. 34 (1991) 361-383.
47. R. Bulirsch, J. Stoer, Numerical treatment of ordinary differential equations by extrapolation methods, Numer. Math. 8 (1966) 1-13.
48. C. Carstensen, On a general epsilon algorithm, in: C. Brezinski (Ed.), Numerical and Applied Mathematics, Baltzer, Basel, 1989, pp. 437-441.
49. J.L. Chabert et al., Histoire d'Algorithmes, Belin, Paris, 1994.
50. L. Collatz, Numerische Behandlung von Differentialgleichungen, Springer, Berlin, 1951.
51. F. Cordellier, Démonstration algébrique de l'extension de l'identité de Wynn aux tables de Padé non normales, in: L. Wuytack (Ed.), Padé Approximation and its Applications, Lecture Notes in Mathematics, Vol. 765, Springer, Berlin, 1979, pp. 36-60.
52. M. Crouzeix, A.L. Mignot, Analyse Numérique des Equations Différentielles, 2nd Edition, Masson, Paris, 1989.
53. A. Cuyt, L. Wuytack, Nonlinear Methods in Numerical Analysis, North-Holland, Amsterdam, 1987.
54. J.P. Delahaye, Automatic selection of sequence transformations, Math. Comp. 37 (1981) 197-204.
55. J.P. Delahaye, Sequence Transformations, Springer, Berlin, 1988.
56. J.P. Delahaye, B. Germain-Bonne, Résultats négatifs en accélération de la convergence, Numer. Math. 35 (1980) 443-457.
57. J. Dutka, Richardson-extrapolation and Romberg-integration, Historia Math. 11 (1984) 3-21.
58. A. Fdil, Sélection entre procédés d'accélération de la convergence, M2AN 30 (1996) 83-101.
59. A. Fdil, A new technique of selection between sequence transformations, Appl. Numer. Math. 25 (1997) 21-40.
60. W.F. Ford, A. Sidi, An algorithm for a generalization of the Richardson extrapolation process, SIAM J. Numer. Anal. 24 (1987) 1212-1232.
61. E. Gekeler, On the solution of systems of equations by the epsilon algorithm of Wynn, Math. Comp. 26 (1972) 427-436.
62. B. Germain-Bonne, Transformations de suites, RAIRO R1 (1973) 84-90.
63. B. Germain-Bonne, Estimation de la Limite de Suites et Formalisation de Procédés d'Accélération de la Convergence, Thèse d'Etat, Université de Lille I, 1978.
64. J. Gilewicz, Numerical detection of the best Padé approximant and determination of the Fourier coefficients of insufficiently sampled function, in: P.R. Graves-Morris (Ed.), Padé Approximants and their Applications, Academic Press, New York, 1973, pp. 99-103.
65. W.B. Gragg, On extrapolation algorithms for initial-value problems, SIAM J. Numer. Anal. 2 (1965) 384-403.
66. B. Grammaticos, A. Ramani, Integrability - and how to detect it, in: Y. Kosmann-Schwarzbach et al. (Eds.), Integrability of Nonlinear Systems, Springer, Berlin, 1997, pp. 30-94.
67. B. Grammaticos, A. Ramani, V.G. Papageorgiou, Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67 (1991) 1825-1828.
68. P.R. Graves-Morris, Vector valued rational interpolants I, Numer. Math. 42 (1983) 331-348.
69. P.R. Graves-Morris, C.D. Jenkins, Vector valued rational interpolants III, Constr. Approx. 2 (1986) 263-289.
70. H.L. Gray, T.A. Atchison, G.V. McWilliams, Higher order G - transformations, SIAM J. Numer. Anal. 8 (1971) 365-381.
71. T. Håvie, Generalized Neville type extrapolation schemes, BIT 19 (1979) 204-213.
72. H. Hankel, Ueber eine besondere Classe der symmetrischen Determinanten, Inaugural Dissertation, Universität Göttingen, 1861.
73. A. Hirayama, K. Shimodaira, H. Hirose, Takakazu Seki's Collected Works Edited with Explanations, Osaka Kyoiku Tosho, Osaka, 1974.
74. H.H.H. Homeier, A hierarchically consistent, iterative sequence transformation, Numer. Algorithms 8 (1994) 47-81.
75. K. Jbilou, H. Sadok, Some results about vector extrapolation methods and related fixed-point iterations, J. Comput. Appl. Math. 36 (1991) 385-398.
76. K. Jbilou, H. Sadok, Hybrid vector transformations, J. Comput. Appl. Math. 81 (1997) 257-267.
77. D.C. Joyce, Survey of extrapolation processes in numerical analysis, SIAM Rev. 13 (1971) 435-490.
78. K. Kommerell, Das Grenzgebiet der Elementaren und Höheren Mathematik, Verlag Köhler, Leipzig, 1936.
79. M. Kzaz, Gaussian quadrature and acceleration of convergence, Numer. Algorithms 15 (1997) 75-89.
80. M. Kzaz, Convergence acceleration of the Gauss-Laguerre quadrature formula, Appl. Numer. Math. 29 (1999) 201-220.
81. P.J. Laurent, Un théorème de convergence pour le procédé d'extrapolation de Richardson, C.R. Acad. Sci. Paris 256 (1963) 1435-1437.
82. H. Lavastre, On the stochastic acceleration of sequences of random variables, Appl. Numer. Math. 15 (1994) 77-98.
83. H. Le Ferrand, Convergence of the topological ε-algorithm for solving systems of nonlinear equations, Numer. Algorithms 3 (1992) 273-284.
84. H. Le Ferrand, Recherches d'extrema par des méthodes d'extrapolation, C.R. Acad. Sci. Paris, Sér. I 318 (1994) 1043-1046.
85. D. Levin, Development of non-linear transformations for improving convergence of sequences, Int. J. Comput. Math. B3 (1973) 371-388.
86. P.M. Lima, M.P. Carpentier, Asymptotic expansions and numerical approximation of nonlinear degenerate boundary-value problems, Appl. Numer. Math. 30 (1999) 93-111.
87. P. Lima, T. Diogo, An extrapolation method for a Volterra integral equation with weakly singular kernel, Appl. Numer. Math. 24 (1997) 131-148.
88. P.M. Lima, M.M Graça, Convergence acceleration for boundary value problems with singularities using the E-algorithm, J. Comput. Appl. Math. 61 (1995) 139-164.
89. S. Lubkin, A method of summing infinite series, J. Res. Natl. Bur. Standards 48 (1952) 228-254.
90. C. Maclaurin, Treatise of Fluxions, Edinburgh, 1742.
91. G.I. Marchuk, V.V. Shaidurov, Difference Methods and their Extrapolations, Springer, Berlin, 1983.
92. A.C. Matos, Acceleration methods based on convergence tests, Numer. Math. 58 (1990) 329-340.
93. A.C. Matos, Linear difference operators and acceleration methods, IMA J. Numer. Anal., to appear.
94. A.C. Matos, M. Prévost, Acceleration property for the columns of the E-algorithm, Numer. Algorithms 2 (1992) 393-408.
95. J.C. Maxwell, A Treatise on Electricity and Magnetism, Oxford University Press, Oxford, 1873.
96. J.B. McLeod, A note on the ε-algorithm, Computing 7 (1971) 17-24.
97. G. Meinardus, G.D. Taylor, Lower estimates for the error of the best uniform approximation, J. Approx. Theory 16 (1976) 150-161.
98. A. Messaoudi, Recursive interpolation algorithm: a formalism for solving systems of linear equations - I, Direct methods, J. Comput. Appl. Math. 76 (1996) 13-30.
99. A. Messaoudi, Recursive interpolation algorithm: a formalism for solving systems of linear equations - II, Iterative methods, J. Comput. Appl. Math. 76 (1996) 31-53.
100. A. Messaoudi, Matrix extrapolation algorithms, Linear Algebra Appl. 256 (1997) 49-73.
101. R.M. Milne, Extension of Huygens' approximation to a circular arc, Math. Gaz. 2 (1903) 309-311.
102. P. Mortreux, M. Prévost, An acceleration property for the E-algorithm for alternate sequences, Adv. Comput. Math. 5 (1996) 443-482.
103. G. Mühlbach, Neville-Aitken algorithms for interpolating by functions of Čebyšev-systems in the sense of Newton and in a generalized sense of Hermite, in: A.G. Law, B.N. Sahney (Eds.), Theory of Approximation with Applications, Academic Press, New York, 1976, pp. 200-212.
104. H. Naegelsbach, Studien zu Fürstenau's neuer Methode der Darstellung und Berechnung der Wurzeln algebraischer Gleichungen durch Determinanten der Coefficienten, Arch. Math. Phys. 59 (1876) 147-192; 61 (1877) 19-85.
105. H. Naegelsbach, Studien zu Fürstenau's neuer Methode der Darstellung und Berechnung der Wurzeln algebraischer Gleichungen durch Determinanten der Coefficienten, Arch. Math. Phys. 59 (1876) 147-192; 61 (1877) 19-85.
106. A. Nagai, J. Satsuma, Discrete soliton equations and convergence acceleration algorithms, Phys. Lett. A 209 (1995) 305-312.
107. A. Nagai, T. Tokihiro, J. Satsuma, The Toda molecule equation and the ε-algorithm, Math. Comp. 67 (1998) 1565-1575.
108. T.H. O'Beirne, On linear iterative processes and on methods of improving the convergence of certain types of iterated sequences, Technical Report, Torpedo Experimental Establishment, Greenock, May 1947.
109. N. Osada, An acceleration theorem for the ρ-algorithm, Numer. Math. 73 (1996) 521-531.
110. N. Osada, Vector sequence transformations for the acceleration of logarithmic convergence, J. Comput. Appl. Math. 66 (1996) 391-400.
111. K.J. Overholt, Extended Aitken acceleration, BIT 5 (1965) 122-132.
112. V. Papageorgiou, B. Grammaticos, A. Ramani, Integrable difference equations and numerical analysis algorithms, in: D. Levi et al. (Eds.), Symmetries and Integrability of Difference Equations, CRM Proceedings and Lecture Notes, Vol. 9, AMS, Providence, RI, 1996, pp. 269-279.
113. V. Papageorgiou, B. Grammaticos, A. Ramani, Integrable lattices and convergence acceleration algorithms, Phys. Lett. A 179 (1993) 111-115.
114. V. Papageorgiou, B. Grammaticos, A. Ramani, Orthogonal polynomial approach to discrete Lax pairs for initial-boundary value problems of the QD algorithm, Lett. Math. Phys. 34 (1995) 91-101.
115. R. Pennacchi, Le trasformazioni razionali di una successione, Calcolo 5 (1968) 37-50.
116. R. Powell, S.M. Shah, Summability Theory and its Applications, Van Nostrand Reinhold, London, 1972.
117. M. Prévost, Acceleration property for the E-algorithm and an application to the summation of series, Adv. Comput. Math. 2 (1994) 319-341.
118. J.D. Pryce, Numerical Solution of Sturm-Liouville Problems, Clarendon Press, Oxford, 1993.
119. B. Rhanizar, On extrapolation methods in optimization, Appl. Numer. Math. 25 (1997) 485-498.
120. L.F. Richardson, The approximate arithmetical solution by finite difference of physical problems involving differential equations, with an application to the stress in a masonry dam, Philos. Trans. Roy. Soc. London, Ser. A 210 (1910) 307-357.
121. L.F. Richardson, The deferred approach to the limit. I: Single lattice, Philos. Trans. Roy. Soc. London, Ser. A 226 (1927) 299-349.
122. W. Romberg, Vereinfachte numerische Integration, Kgl. Norske Vid. Selsk. Forsk. 28 (1955) 30-36.
123. H. Rutishauser, Ausdehnung des Rombergschen Prinzips, Numer. Math. 5 (1963) 48-54.
124. H. Sadok, Quasilinear vector extrapolation methods, Linear Algebra Appl. 190 (1993) 71-85.
125. A. Salam, Non-commutative extrapolation algorithms, Numer. Algorithms 7 (1994) 225-251.
126. A. Salam, An algebraic approach to the vector ε-algorithm, Numer. Algorithms 11 (1996) 327-337.
127. P.R. Graves-Morris, D.E. Roberts, A. Salam, The epsilon algorithm and related topics, J. Comput. Appl. Math. 122 (2000).
128. J.R. Schmidt, On the numerical solution of linear simultaneous equations by an iterative method, Philos. Mag. 7 (1941) 369-383.
129. C. Schneider, Vereinfachte Rekursionen zur Richardson-Extrapolation in Spezialfällen, Numer. Math. 24 (1975) 177-184.
130. M.N. Senhadji, On condition numbers of some quasi-linear transformations, J. Comput. Appl. Math. 104 (1999) 1-19.
131. D. Shanks, 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, July 1949.
132. D. Shanks, Non linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955) 1-42.
133. W.F. Sheppard, Some quadrature formulas, Proc. London Math. Soc. 32 (1900) 258-277.
134. A. Sidi, Extrapolation vs. projection methods for linear systems of equations, J. Comput. Appl. Math. 22 (1988) 71-88.
135. A. Sidi, On a generalization of the Richardson extrapolation process, Numer. Math. 57 (1990) 365-377.
136. A. Sidi, Further results on convergence and stability of a generalization of the Richardson extrapolation process, BIT 36 (1996) 143-157.
137. A. Sidi, A complete convergence and stability theory for a generalized Richardson extrapolation process, SIAM J. Numer. Anal. 34 (1997) 1761-1778.
138. A. Sidi, D. Levin, Prediction properties of the t-transformation, SIAM J. Numer. Anal. 20 (1983) 589-598.
139. E. Stiefel, Altes und neues über numerische Quadratur, Z. Angew. Math. Mech. 41 (1961) 408-413.
140. M. Toda, Waves in nonlinear lattice, Prog. Theor. Phys. Suppl. 45 (1970) 174-200.
141. J. van Iseghem, Convergence of vectorial sequences, applications, Numer. Math. 68 (1994) 549-562.
142. D. Vekemans, Algorithm for the E-prediction, J. Comput. Appl. Math. 85 (1997) 181-202.
143. P. Verlinden, Acceleration of Gauss-Legendre quadrature for an integral with an endpoint singularity, J. Comput. Appl. Math. 77 (1997) 277-287.
144. G. Walz, The history of extrapolation methods in numerical analysis, Report No. 130, Universität Mannheim, Fakultät für Mathematik und Informatik, 1991.
145. G. Walz, Asymptotics and Extrapolation, Akademie, Berlin, 1996.
146. E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Rep. 10 (1989) 189-371.
147. J. Wimp, Sequence Transformations and their Applications, Academic Press, New York, 1981.
148. n) transformation, MTAC 10 (1956) 91-96.
149. P. Wynn, On a procrustean technique for the numerical transformation of slowly convergent sequences and series, Proc. Cambridge Philos. Soc. 52 (1956) 663-671.
150. P. Wynn, Confluent forms of certain non-linear algorithms, Arch. Math. 11 (1960) 223-234.
151. P. Wynn, Acceleration techniques for iterated vector and matrix problems, Math. Comput. 16 (1962) 301-322.
152. P. Wynn, Singular rules for certain non-linear algorithms, BIT 3 (1963) 175-195.
153. P. Wynn, Partial differential equations associated with certain non-linear algorithms, ZAMP 15 (1964) 273-289.
154. P. Wynn, Upon systems of recursions which obtain among the quotients of the Padé table, Numer. Math. 8 (1966) 264-269.
155. P. Wynn, On the convergence and stability of the epsilon algorithm, SIAM J. Numer. Anal. 3 (1966) 91-122.
156. M. Redivo Zaglia, Particular rules for the θ-algorithm, Numer. Algorithms 3 (1992) 353-370.