## 1 Introduction
In Section 4, we present this new algorithm, which is based on the analysis in Section 2 of [17]. In Section 5, we show that, while the results of both the new algorithm and the primal affine algorithm look good on the macro level, there are differences on the micro level. Specifically, the results of the new algorithm are accurate on the micro level while those of the primal affine algorithm are occasionally only approximate. Finally, in Section 6, we summarize the results presented in the previous sections and point out the potential for future algorithms for locally calculated univariate \(L_{1}\) approximating splines and locally calculated bivariate \(L_{1}\) interpolating and approximating splines. All of the quantities in this paper are real quantities. The nodes \(x_{i}\), \(i=0,1,\ldots,I\), are a strictly monotonic but otherwise arbitrary partition of the finite interval \([x_{0},x_{I}]\). Let \(h_{i}=x_{i+1}-x_{i}\). At each node \(x_{i}\), the function value \(z_{i}\) is given, \(i=0,1,\ldots,I\). The slope of the line segment connecting \((x_{i},z_{i})\) and \((x_{i+1},z_{i+1})\) is  
\[\triangle z_{i}:=\frac{z_{i+1}-z_{i}}{h_{i}},\quad i=0,1,\ldots,I-1\,. \tag{1}\]  
The local and global cubic \(L_{1}\) splines discussed in this paper are cubic polynomials in each interval \((x_{i},x_{i+1})\), \(i=0,1,\ldots,I-1\), and are \(C^{1}\) continuous at the nodes. The first derivative of the spline at node \(x_{i}\), \(i=0,1,\ldots,I\), is denoted by \(b_{i}\) (to be determined by minimization of the \(L_{1}\) spline functional). We use \(\delta_{i}\) to denote the slope of the chord between neighboring points:  
\[ \begin{split}\delta_{i}&=\frac{z_{i+1}-z_{i-1}}{x_{ i+1}-x_{i-1}},\quad i=1,2,\ldots,I-1,\\ \delta_{0}&=\frac{z_{1}-z_{0}}{h_{0}}\quad\text{ and } \quad\delta_{I}=\frac{z_{I}-z_{I-1}}{h_{I-1}}\,.\end{split}  \tag{2}\]  
We use \(\zeta\) to denote the linear spline:  
\[\zeta(x)=\frac{(x_{i+1}-x)z_{i}+(x-x_{i})z_{i+1}}{h_{i}},\quad x\in[x_{i},x_{i+ 1}],\quad i=0,1,\ldots,I-1\,. \tag{3}\]  
For the interpolation problem under consideration in the present paper, the nodes and the function values at the nodes are given. Minimization of a spline functional means \"determination of the first derivatives that yield the minimum.\" The first derivatives determined from the minimization along with the fixed nodal and functional values yield the piecewise cubic \(L_{1}\) spline by the standard Hermite interpolation formula. For reference, we present here the spline functionals that define traditional, globally calculated \(L_{1}\) splines (\"global \(L_{1}\) splines\"). Second-derivative-based, first-derivative-based, function-value-based and antiderivative-based cubic \(L_{1}\) splines are calculated by minimizing  
\[\sum_{i=0}^{I-1}\int_{x_{i}}^{x_{i+1}}\left|\frac{\text{d}^{2}z}{\text{d}x^{ 2}}\right|\,\text{d}x\,, \tag{4}\]  
\[\sum_{i=0}^{I-1}\int_{x_{i}}^{x_{i+1}}\frac{1}{h_{i}}\left|\frac{\text{d}z}{ \text{d}x}-\frac{\text{d}\zeta}{\text{d}x}\right|\,\text{d}x\,, \tag{5}\]  
\[\sum_{i=0}^{I-1}\int_{x_{i}}^{x_{i+1}}\frac{1}{h_{i}^{2}}\left|z-\zeta\right| \,\text{d}x \tag{6}\]  
and  
\[\sum_{i=0}^{I-1}\int_{x_{i}}^{x_{i+1}}\frac{1}{h_{i}^{3}}\left|\int_{(x_{i}+x _{i+1})/2}^{x}(z(\xi)-\zeta(\xi))\text{d}\xi\right|\,\text{d}x\,, \tag{7}\]  
respectively, over the finite-dimensional spline space of \(C^{1}\) piecewise cubic polynomials \(z\) that interpolate the data. The \(L_{1}\) splines that minimize these functionals can be nonunique, a situation that will be handled by adding \"regularization terms\" to the functionals when they are minimized by the primal affine algorithm or, in the new algorithm proposed in this paper, by applying a \"choice procedure\" as described below in Section 4. Second-derivative-based \(L_{1}\) splines are the \(L_{1}\) splines commonly encountered in the literature. First-derivative-based \(L_{1}\) splines have been investigated in [10, 11]. Function-value-based \(L_{1}\) splines have been treated in [11].