interpolation A simple means of approximating a function f(x) in which the approximation, say p(x), is constructed by requiring that p(x_i) = f(x_i), i = 0,1,2,…,n

Here f(x_i) are given values p(x_i) that fit exactly at the distinct points x_i (compare smoothing). The value of f can be approximated by p(x) for x ≠ x_i. In practice p is often a polynomial, linear and quadratic polynomials providing the simplest examples. In addition the idea can be extended to include matching of p′(x_i) with f ′(x_i); this is Hermite interpolation. The process is also widely used in the construction of many numerical methods, for example in numerical integration and ordinary differential equations. The interpolating polynomial can be represented in many equivalent forms. For example, when the x_i are equally spaced, the forward and backward difference forms (see difference equation) are convenient. More commonly, nonequally spaced x_i give rise to the divided difference form, which incorporates successive differences (f(x_i+1) – f(x_i))/(x_i+1 – x_i), i = 0,1,2,…,n – 1

These are the first divided differences; second divided differences are obtained by a similar differencing process and so on for higher order differences.

INTERPOLATION

The process of inserting additional words in a complete document or instrument in such manner as to alter its intended meaning; the addition of words to a complete document or instrument.

Interpolation is synonymous with interlineation.