In the mathematical field 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.

Monotone cubic Hermite interpolation

Monotone interpolation can be accomplished using cubic Hermite spline with the tangents m_i modified to ensure the monotonicity of the resulting Hermite spline. An algorithm is also available for monotone quintic Hermite interpolation.

Interpolant selection

There are several ways of selecting interpolating tangents for each data point. This section will outline the use of the Fritsch–Carlson method. Note that only one pass of the algorithm is required. Let the data points be (x_k,y_k) indexed in sorted order for.

Cubic interpolation

After the preprocessing above, evaluation of the interpolated spline is equivalent to cubic Hermite spline, using the data x_k, y_k, and m_k for. To evaluate at x, find the index k in the sequence where x, lies between x_k, and x_{k+1}, that is:. Calculate then the interpolated value is where h_{ii} are the basis functions for the cubic Hermite spline.

Example implementation

The following JavaScript implementation takes a data set and produces a monotone cubic spline interpolant function: