Hermite Cubic Interpolation¶
Definition¶
In Hermite Interpolation, the derivatives as well as the values of the interpolating function are specified at the data points. Specifying derivative values adds more equations to the linear system, and also allows for the derivatives to be smooth.
To have a well-defined solution, the number of equations and the number of parameters to be determined must be equal.
To provide flexibility while maintaining simplicity and computational efficiency, piecewise cubic polynomials are the most common choice for Hermite interpolation.
Parameters¶
A Hermite cubic interpolant is a cubic polynomial interpolant with a continuous first derivative.
A piecewise cubic polynomial with \(n\) knots has \(4(n-1)\) parameters to be determined, since there are \(n-1\) different cubics, and each cubic has 4 coefficients to determine.
Interpolating the given data gives \(2(n-1)\) equations, because each of the \(n-1\) cubics must match the two data points at either end of its subinterval.
Requiring the derivative to be continuous gives \(n-2\) additional equations, because the derivatives of the cubics on either side must match at each of the \(n-2\) interior data points.
Therefore, we have \(3n-4\) equations, which leaves \(n\) free parameters to be determined. Thus, a Hermite cubic interpolant is not unique, and must have the rest of the \(n\) parameters to be determined.
The \(n\) free parameters are thus chosen for many reasons: visually pleasing, satisfying other constraints like monotonicity or convexity.
Note that we can't have more equations than parameters, since then we obtain an overdetermined system, which is approximation and not interpolation anymore.