Homogeneous recurrence relations exhibit a highly numerical unstable behaviour in step-by-step evaluation of succesive terms. It is pointed out that this is a result of the presence of vanishing solutions, which are always added to initial values for the recursion scheme, due to finite machine accuracy. Stabilization of the recursion is shown to be identical with resolving these vanishing contributions with sufficient accuracy. To this end, explicit analytical expresions for these solutions, as products of continued fractions, are given. Application of these vanishing solutions enables us to construct the self-consistent, numerical stable general solution of the recursion relation.
Introductionproblems are related. The instability of the recurrence relations is a reflection of the feature that Quantum mechanical inelastic scattering probthe vanishing solution is always eclipsed by the lems involve an immense number of radial partial exponentially increasing component of the general wave function matrix elements, which differ only solution. In order to solve this apparent problem, I by the quantum number of orbital angulargive an explicit analytical expression for the momentum. Especially in heavy ion collisions, vanishing solution as a product of continued fracthese matrix elements are very cumbersome and tions, which can be evaluated numerically in a even with present-day high-speed computers very stable fashion, and I show how adequate use of hard to calculate [1][2][3][4][5]. Fortunately, many of these this solution stabilizes recursion schemes. The matrix elements are connected by recurrence relamethod described in this paper can be applied tions, so only a few have to be found by explicit very generally, and I will illustrate common feanumerical integration [6,7]. It has, however, been tures with a specific~xamp1e from heavy ion taken for granted that the successive generation of scattering theory. The computer programs were integrals from some initial ones is limited because run on our CDC 175/100 and I used double the relations are highly unstable. A similar probprecision variables with a machine accuracy of 28 lem appears in statistical mechanics on lattices, figures. where the values of microscopic thermodynamic functions on the lattice points are connected by similar relations. With very general arguments, it 2. Instabili~caused by the vanishing solutions can be shown that a recurrence relation has exponentially increasing solutions, except one. TheConsider the homogeneous three-term recurpoint at issue in statistical mechanics is to find this rence relation unique decreasing solution, but due to the intrinsic numerical instability of the recurrence relations,
