Interpolation of off-energy matrices: Symplectic and otherwise (A comparison of methods)
David R. Douglas · 1985
Several tracking programs simulate the motion of off-momentum particles by using transport matrices adjusted to compensate for the variation of the particle rigidity relative to the rigidity of a design particle. The off-momentum matrices are generally determined by one of two methods. First, the matrices may be re-evaluated for exactly the current momentum of the particle. This method is exact and symplectic: no approximations are involved. Alternatively, the matrices are computed for a mesh'' of momentum values; matrices at intermediate values are then obtained by interpolation on momentum. In this note we examine the interpolation procedure used in PATRICIA and show that it does not in general yield a symplectic matrix. We present two alternative interpolation schemes, suggested by Dragt and Leemann, which will yield a symplectic approximate to the exact off-momentum transport matrix. An alternative to symplectic interpolation, that of resymplectification of a nonsymplectic approximate is discussed by Furman in another SSC Technical Memorandum. We conclude with a brief computational comparison of a symplectic and the nonsymplectic interpolation schemes to the benchmark'' of calculating the exact off-energy matrix. We find that the interpolation scheme employed by PATRICIA is only marginally adequate for SSC studies but that a more accurate nonsymplectic interpolation may be sufficiently precise to provide a matrix which is symplectic to within the numerical precision of a computer. Also, we find that the symplectic interpolation algorithm considered does produce the advertised exactly symplectic matrix.