Interpolation Formulas

formulas that give an approximate expression for the function y = f(x) with the help of interpolation, that is, through an interpolation polynomial P_n(x) of degree n, whose values at the given points x₀, x₁ …, x_n coincide with the values y₀, y₁, …, y_n of the function f at these points. The polynomial P_n(x) is uniquely determined, but depending on the problem it is convenient to write it in different forms.

(1) Lagrange’s interpolation formula:

The error introduced by replacing the function f(x) by the expression P_n(x) does not exceed in absolute magnitude

where M is the maximum of the absolute magnitude of the (n + 1) th derivative f^{n + 1} (x) of the function f(x) on the interval [x₀, x_n].

(2) Newton’s interpolation formula: If the points x₀, x₁, …, x_n are situated at equal distances from each other (x_k = x₀ + kh), the polynomial P_n(x) may be written as follows:

(here x₀ + th = x, and Δ^k is the kth order difference: Δ^ky_i = Δ^{k − 1}y_i + _l − Δ^{k − 1}y_i). This is Newton’s formula for forward interpolation; the name of the formula indicates that it contains given values of y corresponding to the nodes of interpolation located only to the right of x_o. This formula is convenient for the interpolation of functions for values of x close to x₀. For interpolation of functions for values of x close to x_n, a similar Newton’s formula is used for backward interpolation. For the interpolation of functions for values of x close to x_k, Newton’s formula is best transformed by an appropriate change of the indexing (see Stirling’s and Bessel’s formulas below).

Newton’s formula can also be written down for unequally spaced nodes by resorting to divided differences. In contrast to Lagrange’s formula, where each term depends on all the nodes of interpolation, any kth term of Newton’s formula depends on the preceding nodes (from the beginning of the indexing), and the addition of new nodes causes only the addition of new terms to the formula (in this lies the advantage of Newton’s formula).

(3) Stirling’s interpolation formula:

Stirling’s formula is used for the interpolation of functions for values of x close to one of the middle nodes a; in this case it is natural to take an odd number of nodes x. _k, …, x _₁, x₀, x₁, …, x_k, considering a as the central node x₀.

(4) Bessel’s interpolation formula:

is used for the interpolation of functions for values of x close to a middle value a between two nodes; here it is natural to take an even number of nodes x_k, …, x₋₁, x₀, x₁, …, x_{k + 1} and to distribute them symmetrically with respect to a(x₀ < a < x_i).