17 E [GeV] p 0 [GeV/c] Bρ [Tm] 5 . 00000000000000 5 . 000510972950664 16 . 67990918220719 Entry ( z = 0 m) Exit ( z = 4 . 8 m) x [cm] 0.0 -0.00449920546655753 p mech x /p 0 0 . 0 4.5494526506306504 × 10 − 10 y [cm] 0.0 0.0 p mech y /p 0 0.0 0.0 ct [cm] 0.0 480.00494875303383 p$_{t}$/p $^{0}$c − 1 . 0000000052213336 -1.0000000052213336 TABLE I: Design Orbit Specifications. 1 . 000056 4 . 800233 × 10 2 0 . 000000 0 . 000000 0 . 000000 − 4 . 500058 × 10 − 3 2 . 235501 × 10 − 7 1 . 000052 0 . 000000 0 . 000000 0 . 000000 − 1 . 001075 × 10 − 9 0 . 000000 0 . 000000 9 . 404373 × 10 − 1 4 . 687866 × 10 2 0 . 000000 0 . 000000 0 . 000000 0 . 000000 − 2 . 465383 × 10 − 4 9 . 404414 × 10 − 1 0 . 000000 0 . 000000 − 4 . 857478 × 10 − 14 − 4 . 499810 × 10 − 3 0 . 000000 0 . 000000 1 . 000000 9 . 897806 × 10 − 3 0 . 000000 0 . 000000 0 . 000000 0 . 000000 0 . 000000 1 . 000000 TABLE II: Linear transfer map R$_{2}$ for the ILC damping ring wiggler. TABLE III: First few nonvanishing Lie generators f$_{m}$ for the ILC damping ring wiggler. FIG. 26: (Color) Illustration of a bent box with straight ends, used for computing a transfer map for bending dipoles with large sagitta. A design orbit is illustrated in red. Index Monomial Coefficient 28 x 3 -1.0738513995490168 × 10 − 9 29 x $^{2}$p$_{x}$ 4.630793805976143 × 10 − 4 33 x $^{2}$p$_{τ}$ 1.1070545309022173 × 10 − 7 34 xp 2 x -2.2214155912078926 × 10 − 1 · · · · · · · · · 209 p 4 τ -4.949353522256519 × 10 − 3 ing all fringe and high-order multipole field effects. In many cases, however, we are interested in magnetic elements with significant sagitta, such as dipoles with large bending angles. In these cases, it is not possible in general to surround the design orbit with a cylindri- cal surface that lies interior to all iron or other magnetic sources. Part II of this paper will describe an alterna- tive, but more computationally intensive, method suit- able for general geometries including that of a bent box with straight ends. See Fig. 26. In this case we obtain simple, geometry-independent kernels for computing the interior vector potential and its derivatives. All the ad- vantages demonstrated in this paper for surface methods will be retained. merically challenging problem whose results are known exactly, we have verified that surface methods have all the advantages claimed in the beginning of Section III. In particular, we demonstrated that errors of a few parts in 10 4 can be achieved for all on-axis gradients C [ n ] $_{m,α}$( z ) required to compute transfer maps through 7 th order, and that the results obtained were remarkably insensi- tive to noise. Moreover, these small errors can be further reduced if desired with the aid of a finer grid. Subsequently we applied surface methods to compute the interior field for the proposed ILC Damping Ring wigglers. Consistent with the accuracy displayed by the monopole-doublet benchmark results, excellent fits were demonstrated for interior fields. We also illustrated the computation of the design orbit and its associated trans- fer map based on surface methods. In summary, the use of surface methods makes it pos- sible, for the first time, to compute for straight beam-line elements realistic transfer maps for real magnets includ- x ! y ! z !