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

DOI:
https://doi.org/10.1090/S0025-5718-1972-0331826-1

MathSciNet review:
0331826

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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 .

**[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 (Unpublished.)**[8]**D. Jordan,*Argonne National Laboratory Library Routine*, ANL C370S--BESJY, October 1967.**[9]**S. Makinouchi, ``Note on the recurrence techniques for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 ,''*Tech. Rep. Osaka Univ.*, v. 15, 1965, pp. 185-201. MR**33**#6813. MR**0198658 (33:6813)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0331826-1

Keywords:
Bessel functions,
difference equations,
error bounds,
FORTRAN,
Miller algorithm,
recursion

Article copyright:
© Copyright 1972
American Mathematical Society