Polynomial Interpolation

Polynomial interpolation constructs the unique polynomial of degree at most n that passes through n+1 given data points, providing a foundation for differentiation, integration, and approximation of functions.

Definition

Polynomial interpolation is the determination of the polynomial of least degree that agrees with prescribed values (and possibly derivatives) at a given set of points, called the interpolation nodes.

Scope

This topic covers the existence and uniqueness of the interpolating polynomial, the Lagrange and Newton divided-difference representations, the barycentric form used for stable evaluation, the interpolation error formula, and the Runge phenomenon that motivates Chebyshev point distributions.

Core questions

• Why is the interpolating polynomial through n+1 distinct points unique, and how is it represented?
• How do the Lagrange and Newton forms compare, and why is the barycentric form preferred for evaluation?
• What does the interpolation error formula say about accuracy, and how does node placement affect it?
• Why does interpolation at equally spaced points fail for high degrees, and how do Chebyshev nodes remedy it?

Key theories

Existence and uniqueness

For n+1 distinct nodes there is exactly one polynomial of degree at most n matching prescribed values, a consequence of the nonsingularity of the Vandermonde system; the Lagrange and Newton forms give two constructive representations of this same polynomial.

Interpolation error and node choice

The interpolation error is the divided difference of order n+1 times the nodal polynomial; minimizing the nodal polynomial's maximum drives the choice of Chebyshev nodes, which suppress the Runge phenomenon and yield near-optimal accuracy.

Mechanisms

The Newton form builds the interpolant incrementally using divided differences, so adding a node requires only one extra term. The barycentric form rewrites the Lagrange interpolant with precomputed weights, allowing the interpolant to be evaluated in linear time per point with excellent numerical stability. The error formula expresses the difference between function and interpolant through a high-order derivative and the product of distances to the nodes, which is small in the interior and large near the ends for equispaced nodes — the source of the Runge phenomenon — but bounded uniformly for Chebyshev nodes.

Clinical relevance

Polynomial interpolation is the building block for numerical differentiation and integration formulas, for constructing quadrature and finite-difference stencils, for spectral methods, and for evaluating tabulated functions; its error analysis informs how densely and where to sample data for accurate reconstruction.

History

Interpolation formulas date to Newton and Lagrange, but the modern understanding was sharpened by Runge's 1901 example showing divergence at equispaced points and by the twentieth-century recognition that Chebyshev nodes and the stable barycentric formula make high-degree interpolation both accurate and practical.

Key figures

• Joseph-Louis Lagrange
• Isaac Newton
• Carl Runge
• Pafnuty Chebyshev

Seminal works

• trefethen2013
• powell1981

Frequently asked questions

Is a higher-degree interpolating polynomial always more accurate?

Not necessarily. With equally spaced nodes, raising the degree can cause large oscillations near the interval ends (the Runge phenomenon) and reduce accuracy. Using Chebyshev-distributed nodes or piecewise (spline) interpolation restores reliable convergence.

Which representation of the interpolant should be used in practice?

The barycentric form is generally preferred: once its weights are computed, it evaluates the interpolant quickly and is numerically stable, unlike directly solving the Vandermonde system, which is ill-conditioned.