Newton’s first interpolation formula

Let in equidistant points , where h – step of interpolation , values are given for function . It is required to pick up polynomial of degree not above n , satisfying to conditions (1).

Let's enter finite differences for sequence of values :

(2)

Conditions (1) are equivalent to the equalities:

at

Lowering the transformations resulted in [1 ], we’ll finally receive Newton’s first interpolation formula :

(3)

where – number of steps of interpolation from beginning point up to point х .

The formula (3) is expedient for using for interpolation of function in locality of beginning point , where q modulo a little.

In special cases we have:

at n = 1 – formula of linear interpolation :

;

at n = 2 – formula of square-law or parabolic interpolation :