Monotone cubic interpolation

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In the mathematical subfield of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated.

Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation.

1 Monotone cubic Hermite interpolation
- 1.1 Data preprocessing
- 1.2 Cubic interpolation
2 References

[edit] Monotone cubic Hermite interpolation

Example showing cubic and monotone cubic interpolation of a monotone data set

Monotone interpolation can be accomplished using cubic Hermite spline with the tangents $m i$ modified to ensure monotoness of the resulting Hermite spline.

[edit] Data preprocessing

Let the data points be $(x k, y k)$ for $k = 1,..., n$

Initialize tangents at every data point,

$m_k = \frac{y_{k}-y_{k-1}}{2(x_{k}-x_{k-1})} + \frac{y_{k+1}-y_k}{2(x_{k+1}-x_k)}$

for $k = 2,..., n - 1$ . For the endpoints, use one-sided differences:

$m_1 = \frac{y_2-y_1}{x_2 - x_1} \quad \text{and} \quad m_n = \frac{y_n - y_{n-1}}{x_n - x_{n-1}}$
Initialize $Δ k = (y k + 1 - y k) / (x k + 1 - x k)$ for $k = 1,..., n - 1$ .
For $k = 1,..., n - 1$ , if $Δ k = 0$ , then set $m k = m k + 1 = 0$ .
Let $α k = m k / Δ k$ and $β k = m k + 1 / Δ k$ .
Now, if α_k + β_k − 2 > 0 and none of the three conditions below are satisfied
1. $2\alpha_k + \beta_k - 3 \leq 0$ ,
2. $\alpha_k + 2\beta_k - 3 \leq 0$ ,
3. $\alpha_k - 1/3 \frac{(2\alpha_k + \beta_k - 3)^2}{\alpha_k + \beta_k -2} \geq 0$ ,

then set

m k = τ k α k Δ k

and

m k + 1 = τ k β k Δ k

where $\tau_k = 3(\alpha_k^2 + \beta_k^2)^{-1/2}$ .

[edit] Cubic interpolation

After the preprocessing, evaluation of the interpolated spline is equivalent to cubic Hermite spline, using the data $x k$ , $y k$ , and $m k$ for $k = 1,..., n$ .

To evaluate at $x$ , find the largest $x lower$ and smallest $x upper$ among $x k$ such that $x_\text{lower} \leq x \leq x_\text{upper}$ . Calculate