Interpolator

An interpolator is a function or algorithm that estimates unknown values within the range of a discrete set of known data points. Given a set of data points (x_i, y_i) with distinct x_i, an interpolator F produces values F(x) for x in the domain, such that F(x_i) = y_i for all i. Interpolation is distinct from extrapolation, which estimates outside the known x-range.

Common interpolation methods include polynomial interpolation, piecewise polynomial interpolation, and non-polynomial approaches. Polynomial interpolation constructs a single polynomial that passes through all data points, using forms such as the Lagrange or Newton representations. While exact, high-degree polynomials can exhibit oscillations and numerical instability, especially with many points (the Runge phenomenon).

Piecewise polynomial methods fit low-degree polynomials on subintervals and stitch them together to form a smooth global curve. Splines, particularly cubic splines, are widely used for their smoothness and stability, providing continuous derivatives at the join points. Linear interpolation connects neighboring points with straight lines, offering simplicity and locality. Monotone and shape-preserving variants aim to maintain monotonicity or other qualitative features of the data.

Other approaches include radial basis function interpolation, which uses distance-based kernels suitable for multidimensional data, and nearest-neighbor interpolation, which assigns the value of the closest data point and is computationally simple but less accurate.

Applications of interpolation span data visualization, numerical simulation, geographic information systems, computer graphics, and signal processing. An interpolant is exact at the known data points, and its accuracy depends on data distribution, the smoothness of the underlying function, and the chosen method.