# Interpolation Formulas

## 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 *x*_{0}, *x*_{1} …, *x*_{n} coincide with the values *y*_{0}, *y*_{1}, …, *y _{n}* of the function

*f*at these points. The polynomial

*P*is uniquely determined, but depending on the problem it is convenient to write it in different forms.

_{n}(x)(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 [x_{0}, x_{n}].

(2) Newton’s interpolation formula: If the points *x*_{0}, *x*_{1}, …, *x*_{n} are situated at equal distances from each other (*x*_{k} = *x*_{0} + *kh*), the polynomial *P _{n}(x)* may be written as follows:

(here *x*_{0} + *th* = *x*, and Δ^{k} is the *k*th order difference: Δ^{k}*y*_{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 *x*_{0}. 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*, Newton’s formula is best transformed by an appropriate change of the indexing (see Stirling’s and Bessel’s formulas below).

_{k}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 *k*th 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}, *x*_{0}, *x*_{1}, …, *x*_{k}, considering *a* as the central node *x*_{0}.

(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},

*x*

_{0},

*x*

_{1}, …,

*x*

_{k + 1}and to distribute them symmetrically with respect to

*a*(

*x*

_{0}<

*a*<

*x*

_{i}).

B. I. BITIUTSKOV