Interpolation of functions  Newton’s first interpolation formula
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 squarelaw or parabolic interpolation:
