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Numerical comparisons of nonlinear convergence accelerators


Authors: David A. Smith and William F. Ford
Journal: Math. Comp. 38 (1982), 481-499
MSC: Primary 65B10; Secondary 65-04
DOI: https://doi.org/10.1090/S0025-5718-1982-0645665-1
MathSciNet review: 645665
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Abstract | References | Similar Articles | Additional Information

Abstract: As part of a continuing program of numerical tests of convergence accelerators, we have compared the iterated Aitken's $ {\Delta ^2}$ method, Wynn's $ \varepsilon $ algorithm, Brezinski's $ \theta $ algorithm, and Levin's u transform on a broad range of test problems: linearly convergence alternating, monotone, and irregular-sign series, logarithmically convergent series, power method and Bernoulli method sequences, alternating and monotone asymptotic series, and some perturbation series arising in applications. In each category either the $ \varepsilon $ algorithm or the u transform gives the best results of the four methods tested. In some cases differences among methods are slight, and in others they are quite striking.


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  • [1] M. Abramowitz &. I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nat. Bur. Standards Appl. Math. Series No. 55, Superintendent of Documents, U. S. Government Printing Office, Washington, D. C., 1964. MR 0167642 (29:4914)
  • [2] R. Bellman &. R. Kalaba, "A note on nonlinear summability techniques in invariant imbedding," J. Math. Anal. Appl., v. 6, 1963, pp. 465-472. MR 0148431 (26:5938)
  • [3] C. M. Bender &. T. T. Wu, "Anharmonic oscillator," Phys. Rev., v. 184, 1969, pp. 1231-1260. MR 0260323 (41:4951)
  • [4] W. G. Bickley &. J. C. P. Miller, "The numerical summation of slowly convergent series of positive terms," Philos. Mag., 7th Ser., v. 22, 1936, pp. 754-767.
  • [5] E. Bodewig, "A practical refutation of the iteration method for the algebraic eigenvalue problem," MTAC, v. 8, 1954, pp. 237-240. MR 0064479 (16:288d)
  • [6] C. Brezinski, "Accélération de suites à convergence logarithmique," C.R. Acad. Sci. Paris Sér. A-B, v. 273, 1971, pp. A727-A730.
  • [7] C. Brezinski, "Computation of the eigenelements of a matrix by the $ \varepsilon $-algorithm," Linear Algebra Appl., v. 11, 1975, pp. 7-20. MR 0371044 (51:7265)
  • [8] E. Gekeler, "On the solution of systems of equations by the epsilon algorithm of Wynn," Math. Comp., v. 26, 1972, pp. 427-436. MR 0314226 (47:2778)
  • [9] S. Graffi, V. Grecchi &. B. Simon, "Borel summability: Application to the anharmonic oscillator," Phys. Lett. B, v. 32, 1970, pp. 631-634. MR 0332068 (48:10395)
  • [10] R. T. Gregory &. D. L. Karney, A Collection of Matrices for Testing Computational Algorithms, Wiley-Interscience, New York, 1969. MR 0253538 (40:6752)
  • [11] P. Henrici, Elements of Numerical Analysis, Wiley, New York, 1964. MR 0166900 (29:4173)
  • [12] K. Iguchi, "On the Aitken's $ {\delta ^2}$-process," Inform. Process. in Japan, v. 15, 1975, pp. 36-40.
  • [13] K. Iguchi, "An algorithm of an acceleration process covering the Aitken's $ {\delta ^2}$-process," Inform. Process, in Japan, v. 16, 1976, pp. 89-93. MR 0455267 (56:13506)
  • [14] D. Levin, "Development of non-linear transformations for improving convergence of sequences," Internat. J. Comput. Math., v. 3, 1973, pp. 371-388. MR 0359261 (50:11716)
  • [15] R. S. Martin, C. Reinsch &. J. H. Wilkinson, "Householder's tridiagonalization of a symmetric matrix," Numer. Math., v. 11, 1968, pp. 181-195. MR 1553959
  • [16] A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965. MR 0191070 (32:8479)
  • [17] D. Shanks, "Non-linear transformations of divergent and slowly convergent sequences," J. Math. and Phys., v. 34, 1955, pp. 1-42. MR 0068901 (16:961e)
  • [18] B. Simon, "Coupling constant analyticity for the anharmonic oscillator," Ann. Physics, v. 58, 1970, pp. 76-136. MR 0416322 (54:4397)
  • [19] D. A. Smith &. W. F. Ford, "Acceleration of linear and logarithmic convergence," SIAM J. Numer. Anal., v. 16, 1979, pp. 223-240. MR 526486 (82a:65012)
  • [20] M. Van Dyke, "Analysis and improvement of perturbation series," Quart. J. Mech. Appl. Math., v. 27, 1974, pp. 423-450. MR 0468591 (57:8423)
  • [21] J. H. Wilkinson, "The use of iterative methods for finding the latent roots and vectors of matrices," MTAC, v. 9, 1955, pp. 184-191. MR 0079833 (18:154b)
  • [22] J. Wimp, "Toeplitz arrays, linear sequence transformations, and orthogonal polynomials," Numer. Math., v. 23, 1974, pp. 1-17. MR 0359260 (50:11715)
  • [23] P. Wynn, "On a device for computing the $ {e_m}({S_n})$ transformation," MTAC, v. 10, 1956, pp. 91-96. MR 0084056 (18:801e)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0645665-1
Keywords: Acceleration of convergence, iterated Aitken's $ {\Delta ^2}$, $ \varepsilon $ algorithm, $ \theta $ algorithm, Levin's transforms, linear convergence, logarithmic convergence, power series, Fourier series, power method, Bernoulli's method, asymptotic series, perturbation series, numerical tests
Article copyright: © Copyright 1982 American Mathematical Society

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