Note on backward recurrence algorithms

TL;DR: In this paper, an algorithm is given for the computation of the recessive solution of a second-order linear difference equation, based upon a combination of algorithms due to J.P. Miller and F.W. Olver.

Abstract: An algorithm is given for the computation of the recessive solution of a second- order 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 trunca- tion error. The method is illustrated by application to the well-used example of the Bessel func- tions Jr(x). 1. Introduction and Summary. Let (1) aryr-1 - brYr + CrYr+l = 0 (r = 1, 2, be a given difference equation in which the coefficients a. and cr do not vanish. Suppose that the equation has a pair of solutions fr and gr such that fr/gr -> 0 as r -*> . Then fr is said to be a recessive (or subdominant or distinguished) solution of the difference equation at r = o, and g. is said to be dominant. The recessive solution is unique, apart from a constant factor. The dominant solution is not unique, however, since any constant multiple of fr may be added to gr without affecting the asymptotic form of gr. Computation of fr from (1) by forward recurrence is usually impractical owing to strong instability. On the other hand, backward application of (1) provides a stable way of computing fr (but not g9), since rounding errors grow no faster than the wanted solution, as a rule.* In the next section, we describe briefly two published algorithms which enable fr to be computed without the need for accurate starting values at high values of r. In Section 3, certain difficulties in the implementation of the algorithms are described, and in the next section, it is shown how these difficulties can be overcome by combining the algorithms. In Section 5, the well-used Bessel function example is considered. A computing routine is described in which the truncation error is bounded rigorously, without loss of efficiency. The method is compared with methods of earlier writers. The concluding section, Section 6, gives proofs of certain results used in earlier sections.