Numerical stability in evaluating continued fractions

A careful analysis of the backward recurrence algorithm for evaluating approxi- mants of continued fractions provides rigorous bounds for the accumulated relative error due to rounding. Such errors are produced by machine operations which carry only a fixed number v of significant digits in the computations. The resulting error bounds are expressed in terms of the machine parameter v. The derivation uses a basic assumption about continued fractions, which has played a fundamental role in developing convergence criteria. Hence, its appear- ance in the present context is quite natural. For illustration, the new error bounds are applied to two large classes of continued fractions, which subsume many expansions of special functions of physics and engineering, including those represented by Stieltjes fractions. In many cases, the results insure numerical stability of the backward recurrence algorithm. 1. Introduction. The analytic theory of continued fractions provides a useful means for representation and continuation of special functions of mathematical physics (1), (2), (10). Many applications of continued fractions and the closely related Pade approximants have recently been made in various areas of numerical analysis and of theoretical physics, chemistry and engineering (4), (5), (7). Thus, it is important to establish a sound understanding of the basic computational problems associated with continued fractions. The present paper is written to help fulfill that aim.