Polynomial interpolation theorem

For $n+1$ distinct nodes and assigned values, exactly one polynomial of degree at most $n$ interpolates the data.

Lagrange basis polynomial

$L_j$ equals 1 at node $x_j$ and 0 at every other interpolation node.

Newton interpolation form

A representation built from products of factors that vanish at previously fitted nodes.

The operator norm that measures how strongly interpolation can amplify errors.

Oscillation between nodes that can cause large errors in high-degree interpolation at equally spaced nodes.

Numerical methodsLesson 1 of 2 in the course Interpolation

Polynomial Interpolation: Forms, Error, Node Choice, and Applications

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Summary

Polynomial interpolation constructs a polynomial that passes exactly through a given set of data points. It is a foundational tool for estimating values between measurements and for many numerical algorithms.

Suppose we are given $n+1$ points $(x₀,y₀),\ldots,(xₙ,yₙ)$ , with all the $xᵢ$ distinct. An interpolating polynomial $p$ satisfies $p(xᵢ)=yᵢ$ at every data point. There is always an interpolant of degree at most $n$ .

All flashcards

Polynomial interpolation theorem: For $n+1$ distinct nodes and assigned values, exactly one polynomial of degree at most $n$ interpolates the data.
Lagrange basis polynomial: $L_j$ equals 1 at node $x_j$ and 0 at every other interpolation node.
Newton interpolation form: A representation built from products of factors that vanish at previously fitted nodes.
Lebesgue constant: The operator norm that measures how strongly interpolation can amplify errors.
Runge's phenomenon: Oscillation between nodes that can cause large errors in high-degree interpolation at equally spaced nodes.

Quiz questions

In Newton interpolation form, what are the coefficients $a_j$ ?Answer: Divided differences $[y_0,\ldots,y_j]$ .
When a new data point is added to a Newton interpolant, how can the new term preserve the values at earlier nodes?Answer: It includes factors that vanish at every earlier node.
Under the Lagrange remainder theorem, which quantities determine the interpolation error at a point?Answer: A derivative factor and a product of distances to the nodes.
What is Runge's phenomenon in polynomial interpolation?Answer: Large oscillations between equally spaced nodes as the interpolation degree increases.
For an estimate near the end of an equally spaced data table, which finite-difference formula is generally appropriate?Answer: The backward interpolation formula.

Based on the Wikipedia article Polynomial interpolation, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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