3 A nine-point least squares spline

In the spirit of Burgess & Tully, it is appropriate to seek an optimised least squares spline to the data and that this should now be based on the nine fixed knot positions [x_i=i/4: i=-4,...,+4]. The spline is specified with constant curvature at both ends. The spline evaluation routine takes as input the x value and the spline values at the nine knot positions. The least squares algorithm for optimally determining the knot values follows the same pattern as that of Burgess & Tully. The special cases of under-determined splines, that is when the number of data values $N \leq 9$ were tedious to prepare but use Lagrangian inverse interpolation to reduce progressively the number of active knots to match the number of data values. Figure 2a shows the B-C plot for electron-impact excitation of He I ${1{\rm s}^2}$ ¹S - ${2{\rm s}}$ ${2{\rm p}}$ ¹P. The data are from the assessment by de Heer et al. (1992) and have been analysed as type 1a. Figure 3a shows the He I ${1{\rm s}^2}$ ¹S - ${2{\rm s}}$ ${2{\rm p}}$ ³P electron impact excitation from the same source, analysed as type 3a.

Figure 2: a) Type 1a: B-C nine-knot spline least squares fit to the $\rm HeI(1~^1S - 2~^1P)$dipole-allowed electron-impact collision strength data of de Heer et al. (1992). B=0.593, C=2.251; b) Type 1a: $\Omega /\Omega ^{\rm (approx)}$ ratio plot and interpolating spline for the same dipole-allowed collision strength data. B=0.53828, $b^{\prime }=1.70840$, F1=0.34300, F2=1.62850, F3=1.32310, Xk=2.35670. Note the variability of the tabular data which gives grounds for adjustment of some values. The stateduncertainty of the data is $\sim$10%