why does not "pchip" requires derivative values as input

In my understanding, Piecewise Cubic Hermite polynomials requires both values and derivative values at each point.

Matlab "pchip", https://www.mathworks.com/help/matlab/ref/pchip.html, is exactly Piecewise Cubic Hermite polynomials, but does not require derivatives.

So why no derivatives values are needed as inputs for "pchip"

Answer 1

[Bruno Luong]

Since the derivatives are computed internally as with general polynomial picewise functions.

It satisfies :

• interpolation on data point
• The derivative inside the interval does not change sign (hermit specific), but this constraints are enforced internally

If user imposes the derivarive at the break points then it might not be able to satifsfy the second condition.

To learn more https://en.wikipedia.org/wiki/Monotone_cubic_interpolation

[Xinwei Li]

Hi Bruno,

Many thanks for your explanation. I am not sure i entirely get it.

A follow-up question, could you point me to the more detailed documentation on pchip's internal procedure of computing derivatives?

[Bruno Luong]

Beside my link to Wikipedia, you have 2 references in the MATLAB doc page (the first refence is also cited by wikipedia page)

References

[1] Fritsch, F. N. and R. E. Carlson. "Monotone Piecewise Cubic Interpolation." SIAM Journal on Numerical Analysis. Vol. 17, 1980, pp.238–246.

[2] Kahaner, David, Cleve Moler, Stephen Nash. Numerical Methods and Software. Upper Saddle River, NJ: Prentice Hall, 1988.

[Xinwei Li]

Hi Bruno,

Many thanks for your help. Now I get the idea of internally calculated derivatives, which is by imposing monotonicity.

[Bruno Luong]

"... which is by imposing monotonicity. "

Exactly