ABSTRACT
This chapter shows that "TPSA" or jet calculations from tracking codes are not "analytic" calculations in the traditional sense. The general idea is to extend the operations performed on real numbers by the computer to an algebra acting on truncated power series, as reflected by the name "TPSA". The person most responsible for the introduction of this algebra to beam dynamics is Martin Berz. His original code, the TPSA package later known as the "DA package", was the first automatic differentiation package tailored for beam dynamicists. The chapter examines that the linguistic complementarity and axiomatic systems is related to some philosophical issues about ring dynamics and single-pass systems. In the global/local approach it really does not matter because the analytical theory is based on a judiciously selected set of maps which the code can either compute or help compute by producing the necessary input to an algebraic formula.
