Hermite interpolation

In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of interpolating data points as a polynomial function. The generated Hermite interpolating polynomial is closely related to the Newton polynomial, in that both are derived from the calculation of divided differences. However, the Hermite interpolating polynomial may also be computed without using divided differences, see Chinese remainder theorem § Hermite interpolation.

Unlike Newton interpolation, Hermite interpolation matches an unknown function both in observed value, and the observed value of its first m derivatives. This means that n(m + 1) values

must be known, rather than just the first n values required for Newton interpolation. The resulting polynomial may have degree at most n(m + 1) − 1, whereas the Newton polynomial has maximum degree n − 1. (In the general case, there is no need for m to be a fixed value; that is, some points may have more known derivatives than others. In this case the resulting polynomial may have degree N − 1, with N the number of data points.)

When using divided differences to calculate the Hermite polynomial of a function f, the first step is to copy each point m times. (Here we will consider the simplest case ${\displaystyle m=1}$ for all points.) Therefore, given ${\displaystyle n+1}$ data points ${\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n}}$, and values ${\displaystyle f(x_{0}),f(x_{1}),\ldots ,f(x_{n})}$ and ${\displaystyle f'(x_{0}),f'(x_{1}),\ldots ,f'(x_{n})}$ for a function ${\displaystyle f}$ that we want to interpolate, we create a new dataset

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