Interpolating

Interpolating is the process of estimating unknown values that fall within the range of a set of known data points. Given a collection of samples with coordinates (x_i, y_i), where the x_i are in increasing order, interpolation seeks a function that assigns a value y at any x between these known points, and matches the known y_i exactly within numerical precision. It is distinct from extrapolation, which estimates outside the observed range.

Common methods include linear interpolation, which connects neighboring points with straight lines; polynomial interpolation, which passes a single polynomial through all points (for n+1 points, a degree-n polynomial is determined, using methods such as Lagrange or Newton forms). Polynomial interpolation can be highly sensitive to point distribution and may exhibit oscillations, especially for many points (the Runge phenomenon). Spline interpolation, particularly cubic splines, uses piecewise polynomials joined at knots with continuity conditions on value and derivatives, offering smooth, stable estimates for large data sets. Other approaches include nearest-neighbor and piecewise constant interpolation, or shape-preserving variants that maintain monotonicity or other features of the data.

Key properties include the requirement that the interpolant agrees with the known data at the x_i and the trade-offs among accuracy, smoothness, and computational complexity. Interpolation assumes the underlying function is well-represented by the chosen model over the interpolation domain; inappropriate choices can lead to poor estimates or excessive oscillation.

Applications span numerical analysis, computer graphics and animation, geostatistics and meteorology, time series resampling, and signal processing, where new samples must be inferred from discrete observations without introducing external information.