Interpolation: Overview
Interpolation constructs a function passing exactly through given data points, enabling evaluation, differentiation, and integration between samples.
Filling the gaps between data
Interpolation builds a function that passes exactly through a set of known points. It answers a common question: given measurements at discrete locations, what is a reasonable value in between? Unlike curve fitting, which minimizes error and need not touch the data, an interpolant matches every sample exactly.
Common families
- Polynomial interpolation: a single polynomial through all points (Lagrange, Newton forms).
- Piecewise polynomial: splines join low-degree pieces smoothly, avoiding wild oscillation.
- Trigonometric: for periodic data, using sines and cosines (the basis of Fourier methods).
- Rational: ratios of polynomials, useful for functions with poles.
The existence and uniqueness result
Through any n+1 points with distinct x-values there is exactly one polynomial of degree at most n. This uniqueness means the Lagrange, Newton, and monomial forms all describe the same polynomial; they differ only in how it is written and computed. The choice of form affects numerical stability and the cost of adding new points.
The danger of high degree
A single high-degree polynomial through many equally spaced points can oscillate violently between them, a failure called Runge's phenomenon. This is why piecewise splines or carefully chosen node placements (Chebyshev points) are preferred for large data sets. Low-degree local interpolation is almost always safer than one global polynomial.
Interpolation is pervasive in simulation: reading tabulated material properties, resampling fields between meshes, and reconstructing values at particle positions. The breeder Hyperion models interpolate cross-section and equation-of-state tables that are known only at tabulated points.