A natural [[Spline]] is a [[Spline]] such that the second derivatives of the first and last points are 0. $\huge \begin{align} S''(x_{1}) &= y_{1}'' = 0\\ S''(x_{n}) &= y_{n}'' = 0 \end{align} $ $ \begin{align} S_{i}(x) = &-\frac{1}{6\Delta x_{i}} y_{i}'' (x-x_{i})(x-x_{i+1})(x+x_{i}-2x_{i+1}) \\ &+ \frac{1}{\Delta x_{i}} y_{i+1}'' (x-x_{i})(x-x_{i+1})(x+x_{i+1}-2x_{i}) \\ &+ m_{i} \left( x - \frac{1}{2}(x_{i}+x_{i+1}) \right) + \frac{1}{2}(y_{i}+y_{i+1}) \end{align}$ Following a similar logic shown in [[Clamped Spline#Constraining given points & specific slopes]], we get this [[System of Linear Equations]]: $ \augmented{cccc|c}{ (\Delta x_{0} + \Delta x_{1}) & \Delta x_{1} & 0 &0 & 6\Delta m_{0} \\ \Delta x_{1} & 2(\Delta x_{1}+ \Delta x_{2}) & \Delta x_{2}& 0 & 6\Delta m_{1} \\ 0 & \Delta x_{2} & \ddots & 0 & \vdots \\ & \ddots & \ddots & & 6\Delta m_{n-2} } $