DA Concatenation: Constant Part Ignored

Consider the following two maps in one degree of freedom:

When using an automatic differentiation package such as FPP, one often computes the "optical" properties around a particular orbit, for example, the closed orbit of a circular ring.  In such systems, the actual value of the zero^th order part may be irrelevant. In fact, if we systematically ignore the constant part of polynomials in FPP operations, we do get a true "Differential Algebra." Thus, without going into mathematical details, we refer to DA concatenation any map concatenation in which the constant part of the map is ignored.

Thus, ignoring the constant parts, the concatenation of M₁ and M₂,  M₁ o M₂ is given by:

In FPP, we actually have:

The constant part of M₂ is added "artificially" at the end of the M₁ o M₂ in FPP. The DA concatenation is always a "*" in FPP.

Program for this example