Basic Methods for Computing Special Functions

  • Amparo Gil
  • Javier Segura
  • Nico M. Temme


This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website.


Numerical evaluation of special functions Chebyshev expansions Quadrature methods Transformation of series Continued fractions Asymptotic analysis 

Mathematics Subject Classification (2000)

65D20 41A60 33C05 33C10 33C15 



We thank the referees for their helpful comments and Dr. Ernst Joachim Weniger for providing us with notes that we used for writing Sect. 4.7.2. We acknowledge financial support from Ministerio de Educación y Ciencia, project MTM2006–09050. NMT acknowledges financial support from Gobierno of Navarra, Res. 07/05/2008.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, vol. 55. US Printing Office (1964) Google Scholar
  2. 2.
    Airey, J.R.: The “converging factor” in asymptotic series and the calculation of Bessel, Laguerre and other functions. Philos. Mag. 24, 521–552 (1937) Google Scholar
  3. 3.
    Alhargan, F.A.: Algorithm 804: subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Softw. 26(3), 408–414 (2000) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alhargan, F.A.: Algorithm 855: subroutines for the computation of Mathieu characteristic numbers and their general orders. ACM Trans. Math. Softw. 32(3), 472–484 (2006) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Amos, D.E.: Algorithm 644: a portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Softw. 12(3), 265–273 (1986) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Baker, G.A. Jr.: The theory and application of the Padé approximant method. In: Advances in Theoretical Physics, vol. 1, pp. 1–58. Academic Press, New York (1965) Google Scholar
  7. 7.
    Baker, G.A. Jr.: Essentials of Padé Approximants. Academic Press, New York/London (1975). [A subsidiary of Harcourt Brace Jovanovich, Publishers] MATHGoogle Scholar
  8. 8.
    Baker, G.A. Jr., Graves-Morris, P.: Padé Approximants, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 59. Cambridge University Press, Cambridge (1996) MATHCrossRefGoogle Scholar
  9. 9.
    Baker, L.: C Mathematical Function Handbook. Programming Tools For Engineers and Scientists. McGraw-Hill, New York (1992) Google Scholar
  10. 10.
    Bhattacharya, R., Roy, D., Bhowmick, S.: Rational interpolation using Levin-Weniger transforms. Comput. Phys. Commun. 101(3), 213–222 (1997) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bickley, W.G., Comrie, L.J., Miller, J.C.P., Sadler, D.H., Thompson, A.J.: Bessel Functions. Part II. Functions of Positive Integer Order. British Association for the Advancement of Science, Mathematical Tables, vol. X. University Press, Cambridge (1952) MATHGoogle Scholar
  12. 12.
    Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J.: The SIAM 100-Digit Challenge. SIAM, Philadelphia (2004). A study in high-accuracy numerical computing, With a foreword by David H. Bailey MATHCrossRefGoogle Scholar
  13. 13.
    Boyd, J.P.: The devil’s invention: Asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Math. 56(1), 1–98 (1999) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Brezinski, C.: A Bibliography on Continued Fractions, Padé Approximation, Sequence Transformation and Related Subjects. Prensas Universitarias de Zaragoza, Zaragoza (1991) Google Scholar
  15. 15.
    Brezinski, C.: History of Continued Fractions and Padé Approximants. Springer, Berlin (1991) MATHCrossRefGoogle Scholar
  16. 16.
    Brezinski, C.: Convergence acceleration during the 20th century. J. Comput. Appl. Math. 122(1–2), 1–21 (2000). Numerical analysis 2000, Vol. II: Interpolation and extrapolation MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Brezinski, C., Redivo-Zaglia, M.: Extrapolation Methods. Theory and Practice. Studies in Computational Mathematics, vol. 2. North-Holland, Amsterdam (1991) MATHGoogle Scholar
  18. 18.
    Carlson, B.C.: Special Functions of Applied Mathematics. Academic Press, New York (1977). [Harcourt Brace Jovanovich Publishers] MATHGoogle Scholar
  19. 19.
    Carlson, B.C.: Computing elliptic integrals by duplication. Numer. Math. 33(1), 1–16 (1979) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Carlson, B.C.: Numerical computation of real or complex elliptic integrals. Numer. Algorithms 10(1–2), 13–26 (1995). Special functions (Torino, 1993) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Carlson, B.C., FitzSimons, J.: Reduction theorems for elliptic integrals with the square root of two quadratic factors. J. Comput. Appl. Math. 118(1–2), 71–85 (2000) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Chatterjee, S., Roy, D.: A class of new transforms tailored for the hypergeometric series. Comput. Phys. Commun. 179(8), 555–561 (2008) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Clarkson, P.A., Mansfield, E.L.: The second Painlevé equation, its hierarchy and associated special polynomials. Nonlinearity 16(3), R1–R26 (2003) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Clenshaw, C.W.: A note on the summation of Chebyshev series. Math. Tables Aids Comput. 9(51), 118–120 (1955) MathSciNetMATHGoogle Scholar
  25. 25.
    Clenshaw, C.W.: The numerical solution of linear differential equations in Chebyshev series. Proc. Camb. Philos. Soc. 53, 134–149 (1957) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Clenshaw, C.W.: Chebyshev Series for Mathematical Functions. National Physical Laboratory Mathematical Tables, vol. 5. Her Majesty’s Stationery Office, London (1962). Department of Scientific and Industrial Research MATHGoogle Scholar
  27. 27.
    Cody, W.J.: A survey of practical rational and polynomial approximation of functions. SIAM Rev. 12(3), 400–423 (1970) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Colavecchia, F.D., Gasaneo, G.: f1: a code to compute Appell’s F 1 hypergeometric function. Comput. Phys. Commun. 157(1), 32–38 (2004) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Cuyt, A., Petersen, V.B., Verdonk, B., Waadeland, H., Jones, W.B.: Handbook of Continued Fractions for Special Functions. Springer, New York (2008). With contributions by Franky Backeljauw and Catherine Bonan-Hamada, Verified numerical output by Stefan Becuwe and Cuyt MATHGoogle Scholar
  30. 30.
    Deaño, A., Segura, J., Temme, N.M.: Identifying minimal and dominant solutions for Kummer recursions. Math. Comput. 77(264), 2277–2293 (2008) MATHCrossRefGoogle Scholar
  31. 31.
    DiDonato, A.R., Hershey, A.V.: New formulas for computing incomplete elliptic integrals of the first and second kind. J. Assoc. Comput. Mach. 6, 515–526 (1959) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Dingle, R.B.: Asymptotic expansions and converging factors. I. General theory and basic converging factors. Proc. R. Soc. Lond. Ser. A 244, 456–475 (1958) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Erricolo, D.: Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch’s algorithm. ACM Trans. Math. Softw. 32(4), 622–634 (2006) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Fabijonas, B.R.: Algorithm 838: Airy functions. ACM Trans. Math. Softw. 30(4), 491–501 (2004) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Fabijonas, B.R., Lozier, D.W., Olver, F.W.J.: Computation of complex Airy functions and their zeros using asymptotics and the differential equation. ACM Trans. Math. Softw. 30(4), 471–490 (2004) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Forrey, R.C.: Computing the hypergeometric function. J. Comput. Phys. 137(1), 79–100 (1997) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM Rev. 9(1), 24–82 (1967) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Gautschi, W.: A computational procedure for incomplete gamma functions. ACM Trans. Math. Softw. 5(4), 466–481 (1979) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Gautschi, W.: Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42(1), 110–118 (2002) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2004) MATHGoogle Scholar
  41. 41.
    Gil, A., Segura, J.: Evaluation of Legendre functions of argument greater than one. Comput. Phys. Commun. 105(2–3), 273–283 (1997) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Gil, A., Segura, J.: A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Commun. 108(2–3), 267–278 (1998) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Gil, A., Segura, J.: Evaluation of toroidal harmonics. Comput. Phys. Commun. 124, 104–122 (2000) MATHCrossRefGoogle Scholar
  44. 44.
    Gil, A., Segura, J.: DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Commun. 139(2), 186–191 (2001) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Gil, A., Segura, J.: Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41(3), 827–855 (2003) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Gil, A., Segura, J., Temme, N.M.: Computing toroidal functions for wide ranges of the parameters. J. Comput. Phys. 161(1), 204–217 (2000) MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Gil, A., Segura, J., Temme, N.M.: On nonoscillating integrals for computing inhomogeneous Airy functions. Math. Comput. 70(235), 1183–1194 (2001) MathSciNetMATHGoogle Scholar
  48. 48.
    Gil, A., Segura, J., Temme, N.M.: Algorithm 819: AIZ, BIZ: two Fortran 77 routines for the computation of complex Airy functions. ACM Trans. Math. Softw. 28(3), 325–336 (2002) MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Gil, A., Segura, J., Temme, N.M.: Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions. ACM Trans. Math. Softw. 28(4), 436–447 (2002) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Gil, A., Segura, J., Temme, N.M.: Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30(1), 11–23 (2002) MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Gil, A., Segura, J., Temme, N.M.: Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Softw. 30(2), 159–164 (2004) MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Gil, A., Segura, J., Temme, N.M.: Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Softw. 30(2), 145–158 (2004) MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Gil, A., Segura, J., Temme, N.M.: The ABC of hyper recursions. J. Comput. Appl. Math. 190(1–2), 270–286 (2006) MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. SIAM, Philadelphia (2007) MATHCrossRefGoogle Scholar
  55. 55.
    Gil, A., Segura, J., Temme, N.M.: Computing the conical function \(P^{\mu}_{-1/2+i\tau}(x)\). SIAM J. Sci. Comput. 31(3), 1716–1741 (2009) MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Gil, A., Segura, J., Temme, N.M.: Fast and accurate computation of the Weber parabolic cylinder function w(a,x) (2009). Submitted to IMA J. Numer. Anal. Google Scholar
  57. 57.
    Gil, A., Segura, J., Temme, N.M.: Algorithm 850: Real parabolic cylinder functions U(a,x), V(a,x). ACM Trans. Math. Softw. 32(1), 102–112 (2006) MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Gil, A., Segura, J., Temme, N.M.: Computing the real parabolic cylinder functions U(a,x), V(a,x). ACM Trans. Math. Softw. 32(1), 70–101 (2006) MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Gil, A., Segura, J., Temme, N.M.: Numerically satisfactory solutions of hypergeometric recursions. Math. Comput. 76(259), 1449–1468 (2007) MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rules. Math. Comput. 23(106), 221–230 (1969). Loose microfiche suppl. A1–A10 MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Grad, J., Zakrajšek, E.: Method for evaluation of zeros of Bessel functions. J. Inst. Math. Appl. 11, 57–72 (1973) MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Graffi, S., Grecchi, V.: Borel summability and indeterminacy of the Stieltjes moment problem: Application to the anharmonic oscillators. J. Math. Phys. 19(5), 1002–1006 (1978) MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Graves-Morris, P.R., Roberts, D.E., Salam, A.: The epsilon algorithm and related topics. J. Comput. Appl. Math. 122(1–2), 51–80 (2000). Numerical analysis 2000, vol. II: Interpolation and extrapolation MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Hart, J.F., Cheney, E.W., Lawson, C.L., Maehly, H.J., Mesztenyi, C.K., Rice, J.R., Thacher, H.C. Jr., Witzgall, C.: Computer Approximations. SIAM Ser. in Appl. Math. Wiley, New York (1968) MATHGoogle Scholar
  65. 65.
    Homeier, H.H.H.: Scalar Levin-type sequence transformations. J. Comput. Appl. Math. 122(1–2), 81–147 (2000). Numerical analysis 2000, Vol. II: Interpolation and extrapolation MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Huber, T., Maître, D.: HypExp 2, expanding hypergeometric functions about half-integer parameters. Comput. Phys. Commun. 178(10), 755–776 (2008) MATHCrossRefGoogle Scholar
  67. 67.
    Ikebe, Y.: The zeros of regular Coulomb wave functions and of their derivatives. Math. Comput. 29, 878–887 (1975) MathSciNetMATHGoogle Scholar
  68. 68.
    Inghoff, T., Fritzsche, S., Fricke, B.: Maple procedures for the coupling of angular momenta. IV: Spherical harmonics. Comput. Phys. Commun. 139(3), 297–313 (2001) MATHCrossRefGoogle Scholar
  69. 69.
    Johnson, J.H., Blair, J.M.: REMES2—a Fortran program to calculate rational minimax approximations to a given function. Technical Report AECL-4210, Atomic Energy of Canada Limited. Chalk River Nuclear Laboratories, Chalk River, Ontario (1973) Google Scholar
  70. 70.
    Kodama, M.: Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Softw. 34(4), Art. 22, 21 (2008) Google Scholar
  71. 71.
    Levin, D.: Development of non-linear transformations of improving convergence of sequences. Int. J. Comput. Math. 3, 371–388 (1973) MATHCrossRefGoogle Scholar
  72. 72.
    Linhart, J.M.: Algorithm 885: Computing the logarithm of the normal distribution. ACM Trans. Math. Softw. 35(3), Art. 20 (2008) Google Scholar
  73. 73.
    Lozier, D.W., Olver, F.W.J.: Airy and Bessel functions by parallel integration of ODEs. In: Sincovec, R.F., Keyes, D.E., Leuze, M.R., Petzold, L.R., Reed, D.A. (eds.) Parallel Processing for Scientific Computing. Proceedings of the Sixth SIAM Conference, vol. II, pp. 530–538. SIAM, Philadelphia (1993) Google Scholar
  74. 74.
    Lozier, D.W., Olver, F.W.J.: Numerical evaluation of special functions. In: Mathematics of Computation 1943–1993: A Half-century of Computational Mathematics, Vancouver, BC, 1993. Proc. Sympos. Appl. Math., vol. 48, pp. 79–125. Am. Math. Soc., Providence (1994). Updates are available at CrossRefGoogle Scholar
  75. 75.
    Luke, Y.L.: The Special Functions and Their Approximations II. Mathematics in Science and Engineering, vol. 53. Academic Press, New York (1969) MATHGoogle Scholar
  76. 76.
    Luke, Y.L.: Mathematical Functions and Their Approximations. Academic Press, New York (1975) MATHGoogle Scholar
  77. 77.
    MacLeod, A.J.: An instability problem in Chebyshev expansions for special functions. ACM SigNum Newslett. 28(2), 2–7 (1993) MathSciNetCrossRefGoogle Scholar
  78. 78.
    Maino, G., Menapace, E., Ventura, A.: Computation of parabolic cylinder functions by means of a Tricomi expansion. J. Comput. Phys. 40(2), 294–304 (1981) MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    Meinardus, G.: Approximation of Functions: Theory and Numerical Methods. Springer Tracts in Natural Philosophy, vol. 13. Springer, New York (1967). Expanded translation from the German edition. Translated by Larry L. Schumaker MATHCrossRefGoogle Scholar
  80. 80.
    Michel, N.: Precise Coulomb wave functions for a wide range of complex l, η and z. Comput. Phys. Commun. 176, 232–249 (2007) MATHCrossRefGoogle Scholar
  81. 81.
    Michel, N., Stoitsov, M.V.: Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Commun. 178, 535–551 (2008) MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Morris, A.H. Jr.: NSWC library of mathematical subroutines. Naval Surface Warfare Center, Dahlgren Division, Dahlgren, VA (1993) Google Scholar
  83. 83.
    Lloyd Baluk Moshier, S.: Methods and Programs for Mathematical Functions. Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester (1989) MATHGoogle Scholar
  84. 84.
    Murli, A., Rizzardi, M.: Algorithm 682: Talbot’s method for the Laplace inversion problem. ACM Trans. Math. Softw. 16(2), 158–168 (1990) MATHCrossRefGoogle Scholar
  85. 85.
    Noble, C.J.: Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Commun. 159(1), 55–62 (2004) CrossRefGoogle Scholar
  86. 86.
    Olde Daalhuis, A.B., Olver, F.W.J.: On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40(3), 463–495 (1998) MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Paris, R.B., Wood, A.D.: Stokes phenomenon demystified. Bull. Inst. Math. Appl. 31(1–2), 21–28 (1995) MathSciNetMATHGoogle Scholar
  88. 88.
    Powell, M.J.D.: On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria. Comput. J. 9(4), 404–407 (1967) MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C++. Cambridge University Press, Cambridge (2002). The art of scientific computing, 2nd edition, updated for C++ Google Scholar
  90. 90.
    Rice, J.R.: The Approximation of Functions. Vol. I: Linear Theory. Addison-Wesley, Reading (1964) Google Scholar
  91. 91.
    Rizzardi, M.: A modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace transform. ACM Trans. Math. Softw. 21(4), 347–371 (1995) MathSciNetMATHCrossRefGoogle Scholar
  92. 92.
    Schmelzer, T., Trefethen, L.N.: Computing the gamma function using contour integrals and rational approximations. SIAM J. Numer. Anal. 45(2), 558–571 (2007) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    Schonfelder, J.L.: Chebyshev expansions for the error and related functions. Math. Comput. 32(144), 1232–1240 (1978) MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    Schulten, Z., Anderson, D.G.M., Gordon, R.G.: An algorithm for the evaluation of the complex Airy functions. J. Comput. Phys. 31(1), 60–75 (1979) MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    Schulten, Z., Gordon, R.G., Anderson, D.G.M.: A numerical algorithm for the evaluation of Weber parabolic cylinder functions U(a, x), V(a, x), and W(a, ±x). J. Comput. Phys. 42(2), 213–237 (1981) MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    Seaton, M.J.: Coulomb functions for attractive and repulsive potentials and for positive and negative energies. Comput. Phys. Commun. 146(2), 225–249 (2002) MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    Seaton, M.J.: FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Commun. 146(2), 250–253 (2002) MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Seaton, M.J.: NUMER, a code for Numerov integrations of Coulomb functions. Comput. Phys. Commun. 146(2), 254–260 (2002) MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    Segura, J.: Reliable computation of the zeros of solutions of second order linear ODEs with a fourth order method. SIAM J. Numer. Anal. 48(2), 452–469 (2010) MathSciNetMATHCrossRefGoogle Scholar
  100. 100.
    Segura, J.: The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40(1), 114–133 (2002) MathSciNetMATHCrossRefGoogle Scholar
  101. 101.
    Segura, J., de Córdoba, P.F., Ratis, Yu.L.: A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Commun. 105(2–3), 263–272 (1997) MATHCrossRefGoogle Scholar
  102. 102.
    Segura, J., Gil, A.: Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Commun. 115(1), 69–86 (1998) MathSciNetMATHCrossRefGoogle Scholar
  103. 103.
    Segura, J., Temme, N.M.: Numerically satisfactory solutions of Kummer recurrence relations. Numer. Math. 111(1), 109–119 (2008) MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    Shippony, Z., Read, W.G.: A correction to a highly accurate Voigt function algorithm. JQSRT 78(2), 255 (2003) CrossRefGoogle Scholar
  105. 105.
    Smith, D.M.: Algorithm 814: Fortran 90 software for floating-point multiple precision arithmetic, gamma and related functions. ACM Trans. Math. Softw. 27(4), 377–387 (2001) MATHCrossRefGoogle Scholar
  106. 106.
    Stieltjes, T.-J.: Recherches sur quelques séries semi-convergentes. Ann. Sci. École Norm. Sup. (3) 3, 201–258 (1886) MathSciNetMATHGoogle Scholar
  107. 107.
    Talbot, A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl. 23(1), 97–120 (1979) MathSciNetMATHCrossRefGoogle Scholar
  108. 108.
    Talman, J.D.: NumSBT: A subroutine for calculating spherical Bessel transforms numerically. Comput. Phys. Commun. 180(2), 332–338 (2009) MathSciNetMATHCrossRefGoogle Scholar
  109. 109.
    Temme, N.M.: On the numerical evaluation of the modified Bessel function of the third kind. J. Comput. Phys. 19(3), 324–337 (1975) MathSciNetMATHCrossRefGoogle Scholar
  110. 110.
    Temme, N.M.: An algorithm with ALGOL 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives. J. Comput. Phys. 32, 270–279 (1979) MATHCrossRefGoogle Scholar
  111. 111.
    Temme, N.M.: Special Functions. Wiley, New York (1996). An introduction to the classical functions of mathematical physics MATHCrossRefGoogle Scholar
  112. 112.
    Temme, N.M.: Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15(2), 207–225 (1997) MathSciNetMATHCrossRefGoogle Scholar
  113. 113.
    Thompson, W.J.: An Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners, with Programs in Fortran 90 and Mathematica. Wiley, New York (1997) MATHGoogle Scholar
  114. 114.
    Trefethen, L.N., Weideman, J.A.C., Schmelzer, T.: Talbot quadrature and rational approximations. Technical report, Oxford University Computing Laboratory Numerical Analysis Group (2005) Google Scholar
  115. 115.
    Van Deun, J., Cools, R.: Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions. ACM Trans. Math. Softw. 32(4), 580–596 (2006) MATHCrossRefGoogle Scholar
  116. 116.
    Van Loan, C.: Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, vol. 10. SIAM, Philadelphia (1992) MATHCrossRefGoogle Scholar
  117. 117.
    Wang, Z.X., Guo, D.R.: Special Functions. World Scientific, Teaneck (1989). Translated from the Chinese by Guo and X.J. Xia CrossRefGoogle Scholar
  118. 118.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1944) MATHGoogle Scholar
  119. 119.
    Weideman, J.A.C.: Optimizing Talbot’s contours for the inversion of the Laplace transform. Technical Report NA 05/05, Oxford U. Computing Lab. (2005) Google Scholar
  120. 120.
    Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comput. 76(259), 1341–1356 (2007) MathSciNetMATHCrossRefGoogle Scholar
  121. 121.
    Weniger, E.J.: Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep. 10(5,6), 189–371 (1989) CrossRefGoogle Scholar
  122. 122.
    Weniger, E.J.: On the summation of some divergent hypergeometric series and related perturbation expansions. J. Comput. Appl. Math. 32(1–2), 291–300 (1990). Extrapolation and rational approximation (Luminy, 1989) MathSciNetMATHCrossRefGoogle Scholar
  123. 123.
    Weniger, E.J., Čížek, J.: Rational approximations for the modified Bessel function of the second kind. Comput. Phys. Commun. 59(3), 471–493 (1990) MATHCrossRefGoogle Scholar
  124. 124.
    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) MathSciNetMATHCrossRefGoogle Scholar
  125. 125.
    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
  126. 126.
    Wuytack, L.: Commented bibliography on techniques for computing Padé approximants. In: Padé approximation and its applications, Proc. Conf., Univ. Antwerp, Antwerp, 1979. Lecture Notes in Math., vol. 765, pp. 375–392. Springer, Berlin (1979) CrossRefGoogle Scholar
  127. 127.
    Wynn, P.: On a device for computing the e m(S n) tranformation. Math. Tables Aids Comput. 10, 91–96 (1956) MathSciNetMATHCrossRefGoogle Scholar
  128. 128.
    Wynn, P.: Upon systems of recursions which obtain among the quotients of the Padé table. Numer. Math. 8(3), 264–269 (1966) MathSciNetMATHCrossRefGoogle Scholar
  129. 129.
    Zhang, S., Jin, J.: Computation of Special Functions. Wiley, New York (1996) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y CC. de la ComputaciónETSI Caminos, Universidad de CantabriaSantanderSpain
  2. 2.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain
  3. 3.CWIAmsterdamThe Netherlands

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