Monotone cubic interpolation

In the algebraic subfield of after analysis, banausic cubic departure is a alternative of cubic departure that preserves monotonicity of the abstracts set getting interpolated.

Monotonicity is preserved by beeline departure but not affirmed by cubic interpolation.

Monotone cubic Hermite interpolation

Monotone departure can be able application cubic Hermite spline with the tangents adapted to ensure the monotonicity of the consistent Hermite spline.

Interpolant selection

There are several means of selecting interpolating tangents for anniversary abstracts point. This area will outline the use of the Fritsch–Carlson method.

Let the abstracts credibility be for

Compute the slopes of the secant curve amid alternating points:

for .

Initialize the tangents at every abstracts point as the boilerplate of the secants,

for ; these may be adapted in added steps. For the endpoints, use biased differences:

For , if (if two alternating are equal), again set as the spline abutting these credibility accept to be collapsed to bottle monotonicity. Ignore footfall 4 and 5 for those .

Let and . If or are computed to be beneath than zero, again the ascribe abstracts credibility are not carefully monotone. In such cases, piecewise banausic curves can still be generated by allotment , although all-around austere monotonicity is not possible.

To anticipate overshoot and ensure monotonicity, the function

must accept a amount greater than (or according to, if monotonicity charge not be strict) zero. One simple way to amuse this coercion is to bind the consequence of agent to a amphitheater of ambit 3. That is, if , again set and area .

Note that alone one canyon of the algorithm is required.

Cubic interpolation

After the preprocessing, appraisal of the amid spline is agnate to cubic Hermite spline, application the abstracts , , and for .

To appraise at , acquisition the aboriginal amount beyond than , , and the better amount abate than , , a part of such that . Calculate

and

then the interpolant is

where are the base functions for the cubic Hermite spline.