 0  1e-07 -20 -15 -10 -5  0  5  10  15 20 RMS Value of C [0] 7,c (z) using 12 Seeds z (cm) 2.0 $^{x}$ 10 -7 1.8 $^{x}$ 10 -7 1.6 $^{x}$ 10 -7 1.4 $^{x}$ 10 -7 1.2 $^{x}$ 10 -7 8.0 $^{x}$ 10 -8 6.0 $^{x}$ 10 -8 4.0 $^{x}$ 10 -8 2.0 $^{x}$ 10 -8 !"#$%$!# &’$ 0 5 z (cm) By (kG) 20 15 10 !5 ! 10 ! 15 ! 20 0 50 100 150 200 250 300 350 400 450 500 !""" #"""" z (cm) B y (kG) FIG. 21: Fit to the proposed ILC wiggler vertical field versus z , where x = 0 . 4 cm, y = 0 . 2 cm. The solid line is computed using data on the surface of an elliptical cylinder with x$_{max}$ = 4 . 4 cm, y$_{max}$ = 2 . 4 cm, using the polynomial series for B obtained from (11) or (12). Dots represent numerical data provided by OPERA-3d. FIG. 20: The function C [0] 7 ,c as computed from a mesh con tain- ing random noise (61) at each mesh point. Damping of this noise illustrates the effect of smoothing. (Solid line) Result obtained using a circular cylinder with R = 2 cm. (Dashed line) Result obtained using an elliptical cylinder with semimi- nor axis of 2 cm and a semimajor axis of 4 cm. FIG. 22: Fit to the proposed ILC wiggler vertical field versus x , where y = 0 . 1 cm, z = 104 . 2 cm. The solid line is computed using data on the surface of an elliptical cylinder with x$_{max}$ = 4 . 4 cm, y$_{max}$ = 2 . 4 cm, using the polynomial series for B obtained from (11) or (12). Dots represent numerical data provided by OPERA-3d. 0 1 x (cm) By (kG) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.2185 13.2184 13.2183 13.2182 13.2181 1 3.2180 13.2179 13.2178 13.2177 13.2176 $^{13.2175}$!" #" x B (kG) B$^{y}$ (kG) ( x, y ) = (0 . 4 , 0 . 2) cm along the length of the wiggler. The field data (points) are shown along with computed values (solid line). Note that the fitted field captures the fringe-field behavior. The relative error was found to sat- isfy the bound δ | B$_{data}$ − B$_{fit}$ | / | B |$_{peak}$ ≤ 3 . 5 × 10 − $^{4}$. We observe that this error is comparable to that found for the monopole-doublet benchmark. Presumably it is due to errors in numerical integration, errors in interpolating onto the elliptic cylinder, errors arising from neglecting terms beyond degree 6, etc., as well as possible failure of the OPERA-3d data to be Maxwellian. Fig. 22 illus- trates the horizontal roll-off of the vertical field at y = 0 . 1 cm, z = 104 . 2 cm for 0 ≤ x ≤ 1 cm. Note the discrete jumps in the original data, reflecting the number of digits retained in the output of the numerical computation. De- spite the small variation of B$_{y}$ in x , the fit goes through the interior data. Finally, Fig. 23 illustrates the longi- tudinal field B$_{z}$ , again at ( x, y ) = (0 . 4 , 0 . 2) cm along the wiggler. Note that no information about B$_{z}$ was used to generate this field, since only the component of B normal to the elliptic cylinder surface was used to generate the interior solution. The error for B$_{y}$ on-axis lies in the range 0 . 1-0 . 2 G along the length of the wiggler, increasing slightly near the end poles. A plot of residuals in the plane y = 0 is displayed in Fig. 24. Note that the error is within 0 . 6 G over this region of the x - z plane. The error begins to increase rapidly at about x = 2 cm; this may be due to retaining only terms through degree 6 in the on-axis gradients, or perhaps a finite domain of convergence of the associated power series for B$_{y}$ ( x, y, z ). For all | x | ≤ 2 cm, the peak error is 0 . 3 G. We remark that this peak error amounts to a relative error of less than 2 parts in 10 $^{5}$, which is remarkably small compared to the error for the data of Fig. 21. We expect the error to behave like a harmonic function, and therefore it should grow as one approaches the boundary. Conversely, it should be the smallest on the center line. The observed error follows this pattern. Finally, this phenomenon may also be a factor in the observed increase in the error at and beyond x = 2 cm. 15