## 3 Computational Results for Windowed and Global \(L_{1}\) Splines  
A window with an odd number of points is desirable, since it has a \"middle point\" at which one can use a locally calculated derivative as the derivative of the global interpolant. Conversely, a window with an even number of points is not desirable, because it lacks a middle point. Using the primal affine algorithm described in the previous section, we generated computational results for 5-point-window and 7-point-window \(L_{1}\) splines as well as for global \(L_{1}\) splines. In the literature, there have been reports [10, 11] about \(L_{1}\) splines based on spline functionals involving first derivatives and function values, that is, derivatives of degree lower than the standard second degree. Results in [10, 11] suggest that, for global \(L_{1}\) splines, spline functionals based on the first derivative or function value (that is, on functional (5) or (6)) result in improved shape preservation. Those results emphasized overall global preservation of shape on the macro scale but did not treat local preservation of shape on the fine, interval-to-interval scale. Following up on these prior investigations, we revisit here the comparison of \(L_{1}\) splines based on derivatives of degree 2, 1 and 0, add to the comparison \(L_{1}\) splines based on derivatives of degree \(-1\) (antiderivatives) and add the new dimension of considering the window size (5 points, 7 points or global). For this comparison, we use the primal affine algorithm described in Section 2. Computational results are presented in Figures 2-13. In each figure, we highlight in dashed boxes the intervals \([25,27],\ [35,37]\) and \([45,55]\), where differences among the various splines are most prominent. The second-derivative-based 5-point-window spline of Figure 2 and the first-derivative-based and function-value-based global splines of Figures 7 and 10 preserve linearity in the intervals \([25,27]\), \([35,37]\) and \([45,48]\) and avoid extraneous overshoot, extraneous undershoot and extraneous oscillation in the interval \([48.1,55]\). None of the other nine splines of Figures 2-13 is able to preserve linearity and avoid extraneous overshoot, undershoot and oscillation in all of these intervals. In particular, the splines of Figures 3, 4, 8, 11, 12 and 13 do not preserve linearity well on the intervals \([25,27]\), \([35,37]\) and/or \([45,48]\). The splines of Figures 3, 5, 8 and 11 have extraneous oscillation on the interval \([48.1,50.9]\). The splines of Figures 6, 9 and 12 have undershoot in \([48.1,50.9]\). The differences in the large intervals \([27.3,34.7]\) and \([37.3,44.7]\) is a separate issue that is discussed in the remark below. **Remark** The data points immediately to the right and left of the large intervals \([27.3,34.7]\) and \([37.3,44.7]\) lie on a quadratic and a cubic function, respectively. The second- and first-derivative-based splines of Figures 2-7 approximate both the quadratic and the cubic function. In contrast, the function-value-based and antiderivative-based splines of Figures 8-13 avoid approximating the quadratic function in \([27.3,34.7]\), even though they do approximate the cubic function in \([37.3,44.7]\). In classical approximation theory, the ability to approximate or even reproduce quadratic, cubic and higher-degree functions is a goal. In geometric modeling of irregular data, however, reproducing any function of degree higher than 1 is generally considered disadvantageous because it leads to extraneous oscillation, overshoot or undershoot. It is not yet known how to construct \(L_{1}\) splines that generically avoid approximating functions of degree higher than 1. In the present article, we acknowledge this issue but we do not take it into account when assessing the shape-preservation capabilities of the various types of \(L_{1}\) splines.