Monotone cubic interpolation

Remarks

With $h = x_1 - x_0$ and $t = (x - x_0)/h$, the derivative is $$\frac{p'(x)}{\Delta} = \alpha (1 - t)(1 - 3t) + \beta \, t (3t - 2) + 6 t (1 - t).$$ At $t = 0$ and $t = 1$ this gives $\alpha$ and $\beta$, which proves part 1. For fixed $t$, the right side is affine in $(\alpha, \beta)$, so its minimum over the square $[0, 3]^2$ is at a corner. At the four corners it equals $6t(1 - t)$, $3(1 - t)^2$, $3t^2$ and $3(1 - 2t)^2$, all non-negative, which proves part 2.

The Fritsch–Carlson method (1980) uses this to make a monotone $C^1$ interpolant. It starts from some slope estimates and sets $m_k = m_{k+1} = 0$ on flat intervals. Where $\alpha_k^2 + \beta_k^2 > 9$, it multiplies both slopes by $3/\sqrt{\alpha_k^2 + \beta_k^2}$, which moves $(\alpha_k, \beta_k)$ into the disc of radius $3$, a subset of the square. Without such a limiter, a cubic spline through monotone data can overshoot: for data $0, 0, 1, 1$ at $x = 0, 1, 2, 3$, the natural cubic spline dips below $0$ on $(0, 1)$. Example: for $y = 0, 1$ on $[0, 1]$ with $m_0 = m_1 = 3$, one has $\alpha = \beta = 3$ and $p(x) = 3x - 6x^2 + 4x^3$. Then $p'(x) = 3(1 - 2x)^2 \ge 0$, so $p$ is monotone, with a flat point at $x = 1/2$.

Going deeper

For a student

The condition is sufficient, and the exact region of monotone $(\alpha, \beta)$ found by Fritsch and Carlson is larger than the square. Monotone interpolation trades some smoothness for shape preservation: the result is $C^1$ but in general not $C^2$, unlike a cubic spline. It is the basis of the PCHIP routine in many numerical libraries and of shape-preserving interpolation more generally.