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]**Walter Gautschi,*Computational aspects of three-term recurrence relations*, SIAM Rev.**9**(1967), 24–82. MR**0213062**, https://doi.org/10.1137/1009002**[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**71B**(1967), 111–129. MR**0221789****[4]**John G. Wills,*On the use of recursion relations in the numerical evaluation of spherical Bessel functions and Coulomb functions*, J. Computational Phys.**8**(1971), 162–166. MR**0298890****[5]**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**0229407****[6]**G. N. Watson,*A Treatise on the Theory of Bessel Functions*, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR**0010746****[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 (Unpublished.)**[8]**D. Jordan,*Argonne National Laboratory Library Routine*, ANL C370S--BESJY, October 1967.**[9]**Saburo Makinouchi,*Note on the recurrence techniques for the calculation of Bessel functions 𝐽ᵥ(𝑥)*, Tech. Rep. Osaka Univ.**16**(1965), 185–201. MR**0198658**

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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