Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Note on backward recurrence algorithms

Authors: F. W. J. Olver and D. J. Sookne
Journal: Math. Comp. 26 (1972), 941-947
MSC: Primary 65Q05; Secondary 33A40
MathSciNet review: 0331826
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An algorithm is given for the computation of the recessive solution of a secondorder linear difference equation, based upon a combination of algorithms due to J.C.P. Miller and F.W.J. Olver. A special feature is automatic and rigorous control of truncation error.

The method is illustrated by application to the well-used example of the Bessel functions $ {J_r}(x)$.

References [Enhancements On Off] (What's this?)

  • [1] W. Gautschi, ``Computational aspects of three-term recurrence relations,'' SIAM Rev., v. 9, 1967, pp. 24-82. MR 34 #3927. MR 0213062 (35:3927)
  • [2] British Association for the Advancement of Science, ``Bessel functions. Part II,'' Mathematical Tables, v. 10, Cambridge University Press, Cambridge, 1952.
  • [3] F. W. J. Olver, ``Numerical solution of second-order linear difference equations,'' J. Res. Nat. Bur. Standards Sect. B, v. 71, 1967, pp. 111-129. MR 36 #4841. MR 0221789 (36:4841)
  • [4] J. G. Wills, ``On the use of recursion relations in the numerical evaluation of spherical Bessel functions and Coulomb functions,'' J. Computational Phys., v. 8, 1971, pp. 162-166. MR 0298890 (45:7939)
  • [5] F. W. J. Olver, ``Bounds for the solutions of second-order linear difference equations,'' J. Res. Nat. Bur. Standards Sect. B, v. 71, 1967, pp. 161-166. MR 37 #4981 MR 0229407 (37:4981)
  • [6] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1944. MR 0010746 (6:64a)
  • [7] W. Kahan, ``Note on bounds for generating Bessel functions by recurrence.'' (Unpublished.)
  • [8] D. Jordan, Argonne National Laboratory Library Routine, ANL C370S--BESJY, October 1967.
  • [9] S. Makinouchi, ``Note on the recurrence techniques for the calculation of Bessel functions $ {J_\nu }(x)$,'' Tech. Rep. Osaka Univ., v. 15, 1965, pp. 185-201. MR 33 #6813. MR 0198658 (33:6813)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65Q05, 33A40

Retrieve articles in all journals with MSC: 65Q05, 33A40

Additional Information

Keywords: Bessel functions, difference equations, error bounds, FORTRAN, Miller algorithm, recursion
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society