Advertisement

Numerical Algorithms

, 48:301 | Cite as

Numerical treatment of a twisted tail using extrapolation methods

  • Mikael Slevinsky
  • Hassan Safouhi
Original Paper

Abstract

Highly oscillatory integral, called a twisted tail, is proposed as a challenge in The SIAM 100-digit challenge. A Study in High-Accuracy Numerical Computing, where Drik Laurie developed numerical algorithms based on the use of Aitken’s Δ2-method, complex integration and transformation to a Fourier integral. Another algorithm is developed by Walter Gautschi based on Longman’s method; Newton’s method for solving a nonlinear equation; Gaussian quadrature; and the epsilon algorithm of Wynn for accelerating the convergence of infinite series. In the present work, nonlinear transformations for improving the convergence of oscillatory integrals are applied to the integration of this wildly oscillating function. Specifically, the \(\bar{D}\) transformation and its companion the W algorithm, and the G transformation are all used in the analysis of the integral. A Fortran program is developed employing each of the methods, and accuracies of up to 15 correct digits are reached in double precision.

Keywords

Nonlinear transformations Extrapolation method Numerical integration Oscillatory integrals 

References

  1. 1.
    Brezinski, C.: Algorithmes d’Accélérations de la Convergence. Edition Technip, Paris (1978)Google Scholar
  2. 2.
    Brezinski, C., Redivo-Zaglia, M.: Extrapolation Methods: Theory and Practice. Edition North-Holland, Amsterdam (1991)MATHGoogle Scholar
  3. 3.
    Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10, 189–371 (1989)CrossRefGoogle Scholar
  4. 4.
    Weniger, E.J.: Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations. Comput. Phys. 10, 496–503 (1996)CrossRefGoogle Scholar
  5. 5.
    Weniger, E.J., Kirtman, B.: Extrapolation methods for improving the convergence of oligomer calculations to the infinite chain limit of quasi-one dimensional stereoregular polymers. In: Simos, T.E., Avdelas, G., Vigo-Aguiar, J. (eds.) (Gastherausgeber) Numerical Methods in Physics, Chemistry and Engineering. Comput. Math. Appl. 45, 189–215 (2003) (special issue)Google Scholar
  6. 6.
    Weniger, E.J.: A rational approximant for the digamma function. Numer. Algorithms 33, 499–507 (2003)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Weniger, E.J.: A convergent renormalized strong coupling perturbation expansion for the ground state energy of the quartic, sextic and octic anharmonic oscillator. Ann. Phys. (NY) 246, 133–165 (1996)CrossRefMATHGoogle Scholar
  8. 8.
    Safouhi, H., Hoggan, P.E.: Efficient evaluation of Coulomb integrals: the non-linear D- and \(\bar{D}\)-transformations. J. Phys. A: Math. Gen. 31, 8941–8951 (1998)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Safouhi, H., Hoggan, P.E.: Efficient and rapid evaluation of three-center two electron Coulomb and hybrid integrals using nonlinear transformations. J. Comp. Phys. 155, 331–347 (1999)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Safouhi, H.: The properties of sine, spherical Bessel and reduced Bessel functions for improving convergence of semi-infinite very oscillatory integrals: the evaluation of three-center nuclear attraction integrals over B functions. J. Phys. A: Math. Gen. 34, 2801–2818 (2001)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Safouhi, H.: An extremely efficient approach for accurate and rapid evaluation of three- center two-electron Coulomb and hybrid integral over B functions. J. Phys. A: Math. Gen. 34, 881–902 (2001)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Berlu, L., Safouhi, H., Hoggan, P.: Fast and accurate evaluation of three-center two-electron Coulomb, hybrid and three-center nuclear attraction integrals over Slater type orbitals using the \({S}\overline{D}\) transformation. Int. J. Quantum Chem. 99, 221–235 (2004)CrossRefGoogle Scholar
  13. 13.
    Berlu, L., Safouhi, H.: An extremely efficient and rapid algorithm for a numerical evaluation of three-center nuclear attraction integrals over Slater type functions. J. Phys. A: Math. Gen. 36, 11791–11805 (2003)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Berlu, L., Safouhi, H.: A new algorithm for accurate and fast numerical evaluation of hybrid and three-center two-electron Coulomb integrals over Slater type functions. J. Phys. A: Math. Gen. 36, 11267–11283 (2003)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Safouhi, H.: Numerical treatment of two-center overlap integrals. J. Mol. Mod. 12, 213–220 (2006)CrossRefGoogle Scholar
  16. 16.
    Safouhi, H.: Efficient and rapid numerical evaluation of the two-electron four-center Coulomb integrals using nonlinear transformations and practical properties of sine and Bessel functions. J. Comp. Phys. 176, 1–19 (2002)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Berlu, L., Safouhi, H.: Multicenter two-electron Coulomb and exchange integrals over Slater functions evaluated using a generalized algorithm based on nonlinear transformations. J. Phys. A: Math. Gen. 37, 3393–3410 (2004)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Safouhi, H.: Highly accurate numerical results for three-center nuclear attraction and two-electron Coulomb and exchange integrals over Slater type functions. Int. J. Quantum Chem. 100, 172–183 (2004)CrossRefGoogle Scholar
  19. 19.
    Duret, S., Safouhi, H.: The W algorithm and the \(\bar{D}\) transformation for the numerical evaluation of three-center nuclear attraction integrals. Int. J. Quantum Chem. 107, 1060–1066 (2007)CrossRefGoogle Scholar
  20. 20.
    Davis, P.J., Rabinowitz, P.: Methods of numerical integration. Academic Press, Orlando (1994)Google Scholar
  21. 21.
    Evans, G.: Practical numerical integration. Wileys, Chichester (1993)MATHGoogle Scholar
  22. 22.
    Wynn, P.: On a device for computing the e m(S n) transformation. Math. Tables Aids Comput. 10, 91–96 (1956)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Poincaré, H.: Sur les intégrales irrégulières des équations linéaires. Acta. Math. 8, 295–344 (1886)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Wagon, S., Bornemann, F., Laurie, D., Waldvogel, J.: The SIAM 100-Digit Challenge. A Study in High-Accuracy Numerical Computing. SIAM, Philadelphia (2004)MATHGoogle Scholar
  25. 25.
    Gautschi, W.: The Numerical Evaluation of a Challenging Integral. Manuscript, Purdue University (2005). URL:http://www.cs.purdue.edu/homes/wxg/
  26. 26.
    Wynn, P.: Upon a second confluent form the ε-algorithm. Proc. Glascow Math. Assoc. 5, 160–165 (1962)MathSciNetMATHGoogle Scholar
  27. 27.
    Levin, D., Sidi, A.: Two new classes of non-linear transformations for accelerating the convergence of infinite integrals and series. Appl. Math. Comput. 9, 175–215 (1981)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Sidi, A.: Extrapolation methods for oscillating infinite integrals. J. Inst. Maths. Applics. 26, 1–20 (1980)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Sidi, A.: An algorithm for a special case of a generalization of the Richardson extrapolation process. Numer. Math. 38, 299–307 (1982)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Gray, H.L., Atchison, T.A.: Nonlinear transformation related to the evaluation of improper integrals. I. SIAM J. Numer. Anal. 4, 363–371 (1967)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Gray, H.L., Atchison, T.A., McWilliams, G.V.: Higher order G-transformations. SIAM J. Numer. Anal. 8, 365–381 (1971)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Gray, H.L., Wang, S.: A new method for approximating improper integrals. SIAM J. Numer. Anal. 29, 271–283 (1992)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Faà di Bruno, C.F.: Note sur une nouvelle formule de calcul différentiel. Quarterly J. Pure Appl. Math. 1, 359–360 (1857)Google Scholar
  34. 34.
    Huang, H.-N., Marcantognini, S.A.M., Young, N.J.: Chain rules for higher derivatives. The Mathematical Intelligencer 28, 61–69 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Campus Saint-JeanUniversity of AlbertaEdmontonCanada

Personalised recommendations