interpolatio

Interpolatio is the process of estimating unknown values within a range defined by a discrete set of known data points. It aims to construct a function that matches the data at the known points and provides reasonable estimates for intermediate values.

Common methods include polynomial interpolation, such as Lagrange and Newton forms, which fit a global polynomial through all data points; spline interpolation, which assembles low-degree polynomials on subintervals for smooth curves; and local interpolation, which uses nearby points to form estimates. Each method balances accuracy, smoothness, and computational cost.

Interpolation is distinct from extrapolation, which predicts values outside the observed range and generally carries higher risk of error.

Errors depend on the underlying function’s smoothness and the distribution of known points. For high-degree global polynomials with evenly spaced nodes, Runge’s phenomenon can cause oscillations. Splines and other piecewise approaches reduce such issues and often improve stability.

Applications span numerical analysis, data reconstruction in statistics, computer graphics, geostatistics, and signal processing. Limitations include potential overfitting, sensitivity to node placement, and numerical conditioning issues in some methods.