Abstract
We introduce a novel class of non-linear transformations of sequences that can accelerate both linear and logarithmic convergence and perform the summation of divergent series. Those transformations naturally arise when the ratio of successive forward differences of the original sequence are approximated by rational functions. The transformed sequences are thus expressible in terms of generalized hypergeometric functions, whose parameters depend on the original sequence. We explicitly derive analytical expressions for the lowest order transformations, present an alternative derivation using continued fractions, investigate convergence and accelerative properties, and derive the kernels of the transformations. The lowest order transformation is equivalent to Aitken’s Δ2 process, while, in the following order, the transformation specialized to logarithmic convergence is equivalent to Lubkin-W transformation. Numerical examples that illustrate the power and limitations of iterations or combinations of hypergeometric transformations are also investigated.
Similar content being viewed by others
References
Aitken, A.C.: On Bernoulli’s numerical solution of algebraic equations. Proc. R. Soc. Edinb. 46, 289–305 (1927)
Arteca, G.A., Fernández, F.M., Castro, E.A.: Large Order Perturbation Theory and Summation Methods in Quantum Mechanics. Springer, Berlin (1990)
Bateman, H.: Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1953)
Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, Singapore (1987)
Bender, C.M., Wu, T.T.: Anharmonic oscillator. Phys. Rev. 184, 1231 (1969)
Brezinski, C.: Accélération de suites à convergence logarithmique. CR Acad. Sci. Paris 273A, 727–730 (1971)
Brezinski, C.: Convergence acceleration during the 20th century. J. Comput. Appl. Math. 122, 1–21 (2000). Numerical Analysis in the 20th Century Vol. II:Interpolation and Extrapolation
Brezinski, C., Redivo-Zaglia, M.: The genesis and early developments of Aitken’s process, Shanks’ transformation, the epsilon—algorithm, and related fixed point methods. Numer. Algorithms 80, 11–133 (2019)
Caliceti, E., Meyer-Hermann, M., Ribeca, P., Surzhykov, A., Jentschura, U.D.: From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions. Phys. Rep. 446, 1–96 (2007)
Chang, X.K., He, Y., Hu, X.B., Sun, J.Q., Weniger, E.J.: Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order. Numer. Algorithms 83, 593–627 (2020)
Cordellier, F.: Caractérisation des suites que la première étape du 𝜃-algorithme transforme en suites constantes. C.R. Acad. Sci. Paris Ser. A 284, 389–392 (1977)
Delahaye, J.P., Germain-Bonne, B.: The set of logarithmically convergent sequences cannot be accelerated. SIAM J. Numer. Anal. 19, 840–844 (1982)
Drummond, J.E.: The anharmonic oscillator: complex eigenvalues for the ground state with negative quartic or cubic energy distortion. J. Phys. A Math. Gen. 15, 2321 (1982)
Dyson, F.J.: Divergence of perturbation theory in quantum electrodynamics. Phys. Rev. 85, 631 (1952)
Grotendorst, J., Weniger, E.J., Steinborn, E.O.: Efficient evaluation of infinite-series representations for overlap, two-center nuclear attraction, and coulomb integrals using nonlinear convergence accelerators. Phys. Rev. A 33, 3706 (1986)
Hildebrand, B.F.: Introduction to Numerical Analysis, 2nd edn. McGraw-Hill, New York (1974)
Jonquière, A.: Note sur la série \({\sum }_{n= 1}^{\infty } \frac {x^{n}}{n^{s}}\). Bull. de la Soc. Math. de France 17, 142–152 (1889)
Le Guillou, J.C., Zinn-Justin, J.: Large-Order Behaviour of Perturbation Theory. Elsevier (2012)
Levin, D.: Development of non-linear transformations for improving convergence of sequences. Int. J. Comput. Math. 3, 371–388 (1972)
Lubkin, S.: A method of summing infinite series. J. Res. Nat. Bur. Standards 48, 228–254 (1952)
Mera, H., Pedersen, T.G., Nikolić, B.K.: Nonperturbative quantum physics from low-order perturbation theory. Phys. Rev. Lett. 143001, 115 (2015)
Negele, J.W., Orland, H.: Quantum Many-Particle Systems Perseus Books (1998)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press (2010)
Osada, N.: A convergence acceleration method for some logarithmically convergent sequences. SIAM J. Numer. Anal. 27, 178–189 (1990)
Rockett, A.M., Szüsz, P.: Continued Fractions. World Scientific, Singapore (1992)
Shanks, D.: Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, 1–42 (1955)
Sidi, A.: Practical Extrapolation Methods: Theory and Applications. Cambridge University Press, Cambridge UK (2003)
Michael Trott: The Mathematica Guidebook for Symbolics. Springer Science & Business Media (2007)
Wall, H.S.: Analytic Theory of Continued Fractions. D.van Nostrand, New York (1948)
Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10, 189–371 (1989)
Weniger, E.J.: On the derivation of iterated sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Commun. 64, 19–45 (1991)
Weniger, E.J., Čížek, J., Vinette, F.: The summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic, sextic, and octic anharmonic oscillators using nonlinear sequence transformations. J. Math. Phys. 34(2), 571–609 (1993)
Wynn, P.: On a device for computing the em(Sn) transformation. Mathematical Tables and Other Aids to Computation 10, 91–96 (1956)
Wynn, P.: On a procrustean technique for the numerical transformation of slowly convergent sequences and series. Math. Proc. Camb. Philos. Soc. 52(4), 663–671 (1956)
Acknowledgements
The author would like to thank Prof. Claude Brezinski and the anonymous reviewers for invaluable suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
This research did not receive support from any organization. The author declares that he has no conflict of interest. Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Pepino, R.T. Acceleration of sequences with transformations involving hypergeometric functions. Numer Algor 92, 893–915 (2023). https://doi.org/10.1007/s11075-022-01334-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-022-01334-7