Generalised Truncated Power Series Algebra for Fast Particle Accelerator Transport Maps

New Generalised Truncated Power Series Algebra (TPSA) has been developed for extending, simplifying and optimising the transport maps used by particle accelerator simulation codes. TPSA are intensively used in optics code to describe transport maps of the elements constituting the particle accelerator to any order. Generalised TPSA extend the degrees to inhomogeneous ones, where separate degrees can be specified for each variables and constrained by two total orders, one for canonical variables and one for ordinary variables. This allows to track inhomogeneous planes of the 6D phase space with many extra variables. A complete set of new formulas and data structures have been derived to address the problem of memory consumption required for efficient computation of high order TPSA, including generalised indexing, multiplication and composition of inhomogeneous multivariate polynomials. The implementation has been benchmarked against well established libraries for the common subset with TPSA, and outperforms all of them for supported differential algebra operators on low and high orders, and high number of variables.