PCHIP slope selection

Inputs: An integer $n\ge 2$, strictly increasing real nodes $x_0<\cdots <x_{n-1}$, and real values $y_0,\dots ,y_{n-1}$. All operations here are over the real numbers.

Output: Slopes $m_0,\dots ,m_{n-1}$ and the piecewise cubic Hermite interpolant determined by these nodes, values, and slopes under Piecewise cubic Hermite interpolant.

Put $h_i=x_{i+1}-x_i>0$ and $d_i=(y_{i+1}-y_i)\mathrm{/}{h}_i$ for $0\le i<n-1$. Write $\mathrm{sgn}(r)\in \{-1,0,1\}$ for the sign of a real number.

• If $n=2$, set $m_0=m_1=d_0$.
• If $n\ge 3$, for each interior node $1\le k\le n-2$, set $m_k=0$ when $d_{k-1}{d}_k\le 0$. When $d_{k-1}{d}_k>0$, put $w_1=2h_k+h_{k-1}$, $w_2=h_k+2h_{k-1}$, and $m_k=\frac{w_1+w_2}{w_1\mathrm{/}{d}_{k-1}+w_2\mathrm{/}{d}_k}\mathrm{.}$
• For $n\ge 3$, at the left endpoint compute ${\widehat{m}}_0=\frac{(2h_0+h_1)d_0-h_0{d}_1}{h_0+h_1}\mathrm{.}$ Set $m_0=0$ if $\mathrm{sgn}({\widehat{m}}_0)\mathrm{\ne }\mathrm{sgn}(d_0)$. Otherwise set $m_0=3d_0$ if $d_0{d}_1<0$ and $\mathrm{\mid }{\widehat{m}}_0\mathrm{\mid }>3\mathrm{\mid }{d}_0\mathrm{\mid }$; in all remaining cases set $m_0={\widehat{m}}_0$.
• For $n\ge 3$, at the right endpoint compute ${\widehat{m}}_{n-1}=\frac{(2h_{n-2}+h_{n-3})d_{n-2}-h_{n-2}{d}_{n-3}}{h_{n-2}+h_{n-3}}\mathrm{.}$ Apply the preceding clip and cap rule with $d_0,d_1$ replaced by $d_{n-2},d_{n-3}$ and $m_0,{\widehat{m}}_0$ replaced by $m_{n-1},{\widehat{m}}_{n-1}$.

For $n\ge 3$, the algorithm returns the Hermite pieces from the stated values and selected slopes. The two-node branch is the line through the two input points.