Note on backward recurrence algorithms

Olver, F. W. J.; Sookne, D. J.

doi:10.1090/S0025-5718-1972-0331826-1

Note on backward recurrence algorithms
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by F. W. J. Olver and D. J. Sookne PDF

Math. Comp. 26 (1972), 941-947 Request permission

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

Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, DOI 10.1137/1009002

Mathematical Tables

F. W. J. Olver, Numerical solution of second-order linear difference equations, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 111–129. MR 221789
John G. Wills, On the use of recursion relations in the numerical evaluation of spherical Bessel functions and Coulomb functions, J. Comput. Phys. 8 (1971), 162–166. MR 298890, DOI 10.1016/0021-9991(71)90043-x
F. W. J. Olver, Bounds for the solutions of second-order linear difference equations, J. Res. Nat. Bur. Standards Sect. B 71B (1967), 161–166. MR 229407
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746

Argonne National Laboratory Library Routine

Saburo Makinouchi, Note on the recurrence techniques for the calculation of Bessel functions $J_{v}(x)$, Tech. Rep. Osaka Univ. 16 (1965), 185–201. MR 198658