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 :

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