ABSTRACT
Power series calculations can be carried out with TPSA to an arbitrary order with an arbitrary number of parameters and are thus very useful in tracking codes. Analytical calculations are more complex and must be avoided whenever possible. One of the great virtues of map methods in rings resides in the use of a formalism which integrates linear and nonlinear calculations. The entire theory of rings and quasi-periodic systems has only been glanced at without a serious look at applications. They are several reasons for this lack of completeness. The work of Irwin on Lie methods applied to accelerator modules shows how one can make quantum leaps by using the appropriate tools when faced with a complex design problem.
