Applications of differential algebra to single-particle dynamics in storage rings

Model Independent Analysis of Beam Centroid Dynamics in Accelerators

“…I will not look into this general question here. It suffices to point out that many numerical studies [76,75] support the assumption that a 9-th order Taylor map is adequate for the stability studies in an electron machine. In the LER ring example, a 9-th order Taylor map has been used to generate tracking data and compare with the numerical fits.…”

Zlib: A numerical library for optimal design of truncated power series algebra and map parameterization routines

“…A Vps truncated at an order of Q can be mathematically written as [1] [5] O=(J that is, each component of the Vps is a Tps represented the last section. For example, the Zth component would…”

“…where again iOSIP and j OSIP are One-Step Index Pointers and in the double loop, the inner one is vectorized while the outer one is parallelized. It was this fast computational process that allowed the fast Taylor-map tracking for the SSC to high orders (11-or 12-th order) with a computational speed two orders of magnitude faster than the conventional element-by-element tracking [5].…”

Swamp plots for dynamic aperture studies of PEP-II lattices

“…Such inaccuracy and nonsymplecticity have been controversial in the use of one-turn maps for long-term tracking. However, it is well under stood that the required accuracy and symplecticity depend on circumstances [7]. For the PEP-II lattice dynamic aper ture studies (only 1024 turns), through numerous testing we have concluded that a map of 10 th order in Hamiltonian (9" 1 order in Taylor expansion) is accurate enough and 3 Poisson bracket expansion of the Lie transformation is ac curate and symplectic enough.…”

“…The nPB tracking provides a fac tor of 100 in tracking speed and also offers insight into the underlying physics. Map tracking for proton accelera tors [6] [7] [8] is a different problem since there is no signifi cant synchrotron radiation damping.…”