interpolaties

Interpolaties, or interpolations, in mathematics, are methods and results used to construct a function that exactly passes through a given set of data points. Given pairs (x_i, y_i) with distinct x_i, the interpolating function p(x) satisfies p(x_i) = y_i for all i. Interpolation is distinct from extrapolation, which estimates values outside the known data.

Common approaches include polynomial interpolation, which constructs a single polynomial of degree at most n−1 to fit n points. The Lagrange form and Newton’s divided differences are standard algorithms; the barycentric form is particularly efficient for computation. For large data sets, piecewise methods such as spline interpolation, especially cubic splines, provide smooth interpolants by joining low-degree polynomials on subintervals.

Other methods include trigonometric interpolation, which uses sums of sine and cosine terms to fit periodic data, and rational interpolation, which uses ratios of polynomials. Each method has trade-offs: polynomial interpolation can exhibit oscillations for equispaced nodes (the Runge phenomenon), while splines offer local control and reduced oscillations. Error analysis links the interpolation error to the smoothness of the target function, the degree of the interpolant, and the placement of nodes.

Applications span numerical analysis, data fitting, computer graphics, geospatial modeling, and signal processing. In practice, the choice of interpolant reflects a balance between accuracy, smoothness, and computational efficiency, tailored to the nature of the data and the intended use.