Most methods for solving stiff systems are based on implicit formulas and an iteration scheme must be used to solve the, in general, nonlinear equations that arise at each time step. The computational costs of the matrix operations associated with the iteration schemes currently in use often dominate the total cost of the integration. The efficiency of stiff solvers can be improved significantly if these costs can be reduced. A new iteration scheme that automatically partitions the iteration matrix into two parts, one corresponding to transient components and the other corresponding to the smooth components is developed. The resulting structure is then exploited to reduce the cost of updating the iteration matrix whenever a change in the stepsize or order occurs. The convergence of the new scheme is studied and the scheme's applicability to existing stiff solvers is shown. The new scheme is particularly useful in solving large stiff systems that have only a few transient components.