Algorithms for the solution of general systems of stiff differential equations employ integration formulae which are implicit in character. The associated non-linear algebraic equations are solved by an iteration procedure such as the Simplified Newton iteration. In the asymptotic region of integration, the use of predictors involving past values of derivatives leads to inaccurate estimates for the solution of the implicit equations and consequently to unnecessary iterations or even divergence. The reasons for this are analysed and more suitable predictors are proposed. The phenomena are illustrated by simple numerical examples and experience with a number of problems is quoted. For a well-known practical problem and a classical algorithm we have observed an increase in efficiency by a factor of eight.