Interpolieren

Interpolieren (interpolation) is the process of constructing a function that exactly fits a set of known data points and is used to estimate intermediate values. Given a set of pairs (x_i, y_i) with strictly increasing x_i, interpolation yields a function f defined on the interval [x_0, x_m] such that f(x_i) = y_i for each i.

Linear interpolation uses the unique line through two adjacent points to estimate values between them. It is simple and fast but yields limited smoothness. Polynomial interpolation seeks a single polynomial that passes through all data points; it can be written in Lagrange form or Newton form with divided differences. High-degree polynomials can exhibit oscillations and be unstable for many nodes (the Runge phenomenon).

Spline interpolation, especially cubic splines, fits piecewise low-degree polynomials between neighboring nodes with continuity constraints on function values and first and sometimes second derivatives. This approach produces smooth curves with good numerical stability. Other methods include Hermite interpolation, which also matches derivative values, and multivariate approaches such as radial basis function interpolation for higher dimensions.

Interpolation versus extrapolation: interpolation estimates within the data range; extrapolation estimates outside it and is generally more error-prone. The interpolation error depends on the method and the smoothness of the underlying function; for polynomial interpolation, the error term involves the (n+1)st derivative at some point and the product of distances to the nodes. Choosing node positions (e.g., Chebyshev nodes) can reduce maximum error.

Applications include data visualization, numerical analysis, computer graphics, weather and geoscience, and engineering. In practice, the choice of method balances accuracy, stability, and computational cost.