Modern Map Methods for Charged Particle Optics

The Diierential Algebraic Methods, a natural tool for the description and determination of the solution of diierential equations, have proven useful for the computation of aberrations of any desired order in any particle optical system. Besides conventional symplectic systems based on strongly or weakly focusing elements including fringe elds, they also readily allow the treatment of spin dynamics as well as classical synchrotron radiation. In recent years, a variety of codes have been written based on these techniques, including COSY INFINITY, which is currently used by approximately 140 registered users. Several examples of recent applications employing high order methods will be given. While the method makes the problem of computation of Taylor maps straightforward and their manipulation and analysis convenient, for many applications it is important to have exact bounds of the truncation error. Recently it was shown how such information can be determined conveniently and with rather limited eeort. Furthermore, it is often important to know the domain and the speed of convergence of the Taylor expansion. We will show that similar to the conventional DA approach, such information can be obtained by carrying the three elementary operations of addition, multiplication, and diierentiation on the space of innnitely often diierentiable functions to a suitable smaller space that can be described on a computer.