Advances in Numerical Analysis Emphasizing Interval Data

4 Advanced Interpolation

DOI: 10.1201/9781003218173-4

4.1 Hermit Interpolation

In this section, we introduce an interpolation that operates in a multicriteria manner. In this interpolator, we assume that at the interpolation points, the value of the function $f$ is equal to the value of the interpolation function, as well as the value of the finite derivatives of $f$ is equal to the value of the finite derivatives of the interpolation function, and this is the reason for smoothness and accuracy of the interpolation function.

Suppose that $x_i$ s are real numbers for $i=0,\dots ,n$, so that:

$$x_0\le x_i\le \cdots \le x_n$$

Therefore, for $i=0,\dots ,m$, there are points like ${\xi }_i$ with a new arrangement so that:

$${\xi }_0<{\xi }_1<\cdots <{\xi }_m$$

So, we consider the interpolation points as follows:

$$($$