Lagrange Polynomial Interpolation: Construction, Evaluation, and Error

All flashcards

What does polynomial interpolation do?

It constructs a polynomial that matches prescribed values at specified input nodes.

Why is the interpolating polynomial unique for distinct nodes?

The difference of two candidates would have more distinct roots than its degree permits, so it must be zero.

What is the defining property of the Lagrange basis polynomial $\ell_j$?

It equals one at node $x_j$ and zero at every other interpolation node.

What is the interpolation remainder?

The difference $R(x)=f(x)-L(x)$ between the function and its interpolating polynomial.

What does Hermite interpolation add to ordinary interpolation?

It can incorporate derivative data as well as function values.

Quiz questions

1. For distinct nodes $x_0,\ldots,x_k$, what values does the Lagrange basis polynomial $\ell_j$ take at the nodes?Answer: It is one at $x_j$ and zero at every other node.
2. Which expression gives the second barycentric form of the interpolating polynomial?Answer: $\frac{\sum_j w_j y_j/(x-x_j)}{\sum_j w_j/(x-x_j)}$
3. A second-barycentric-form calculation is requested exactly at an interpolation node $x_j$. What should a numerical implementation return?Answer: The stored data value $y_j$.
4. For a sufficiently smooth function $f$ interpolated at distinct nodes $x_0,\ldots,x_k$, which expression gives the classical remainder at a point $x$?Answer: $\frac{f^{(k+1)}(\xi)}{(k+1)!}\prod_{i=0}^{k}(x-x_i)$ for some intermediate $\xi$
5. What behavior is called Runge’s phenomenon in high-degree interpolation with equally spaced nodes?Answer: Large oscillations, especially near the ends of the interval.