Hermite interpolation

So far we have developed approximation methods which make use of function values only. Sometimes, the derivative information may also be available. Methods which make use of derivatives also to construct the approximation are known are Hermite approximations. Here we will look at interpolating given function and derivative information.

1Lagrange form of interpolation¶

Given $(x_i,y_i,y_i')$, $i=0,1,2,\ldots,N$, find a polynomial $p(x)$ such that

There are $2N+2$ conditions given so we can try to find a polynomial of degree $2N+1$ that interpolates the given data. Define

then the Lagrange polynomials are

Define

Since $\ell_i \in \poly_N$ both $r_i$, $s_i$ are in $\poly_{2N+1}$. Since $\ell_i(x_j) = \delta_{ij}$, we have

Then the interpolating polynomial is given by

1.1Uniqueness¶

Suppose there are two polynomials $H(x)$, $G(x)$ of degree $\le 2N+1$ that interpolate the same set of data. Then define

which has degree $\le 2N+1$ and

Thus $R(x)$ has $N+1$ double roots so that, and hence atleast $2N+2$ roots, which implies that $R(x) \equiv 0$.

2Newton form of interpolation¶

The polynomial interpolating $f(x)$ at nodes $z_0,\ldots,z_{2N+1}$ is

with error formula

We can let the nodes coincide and Newtons formula will still make sense.

which gives

This is a polynomial of degree $\le 2N+1$ with error formula

2.1Claim: $p_{2N+1} = H_N$¶

Assume that $f \in \cts^{2N+3}$. Due to the Newton form, we have

Differentiating the error formula

we see that

We have degree $p_{2N+1} \le 2N+1$ and it satisfies the given data $\{x_i,y_i,y_i'\}$, $i=0,1,\ldots,N$. By uniqueness of the Hermite interpolating polynomial, we have $p_{2N+1} = H_N$.