[Feature] Map Inversion of Taylor Series

[MaximeJo]

Hi all,

As Hipparchus has Taylor expansion capacities embedded in the DerivativeStructure class, it would be nice to be able to invert the Taylor expansion and do what it is often called a Taylor map inversion or polynomial partial inversion.

An example of usage in Orekit (which depends on Hipparchus) would be:

• Given an initial state X0 at t0 and a propagation model, compute a first-guess orbital maneuver parameterized with (a,b,c) that leads to a final state Xf at tf
• Obtain the Taylor expansion of Xf depending on the maneuver's parameters: T(Xf) = Taylor(a, b, c)
• Invert the Taylor algebra polynomial to get (a, b, c) = T^(-1)(Xf) as a Taylor expansion.
• Compute (a, b, c) that optimize the maneuver to reach a target state Xt at tf.

Here's a link to a mathematical method used in another library.

[Serrof]

Hi Maxime,

I strongly support the addition of this feature. This algorithm can be found in the Chapter 2 of Modern Map Methods in Particle Beam Physics by Martin Berz (Academic Press 1999). Also, be careful with notations: one can only invert a square map (as many functions than variables in the Taylor algebra), but trivial components can be added for that purpose. A univariate (in that case, this is also called series reversion) example in astrodynamics is Kepler's equation (see https://darioizzo.github.io/audi/notebooks/map_inversion.html).

[Serrof]

There are many ways to validate the implementation of this, for instance:

• Inverse of 1D map on any intrinsic function e.g. expwill be equivalent to directly calling their reciprocal logas a DerivativeStructure
• Any inverted map, if composed ("rebased" as per Hipparchus' nomenclature) with the original one, will give the so-called identity map.

Luc if you have any interrogations on the process described in Maxime's reference, don't hesitate to ask me. But I reckon that it is rather straightforward compared to more complicated things you handled fine before!