Interpolation of functions - Introduction
Interpolation is an operation of approach of function given in separate points inside of some given interval. The elementary problem of interpolation consists in the following. On a segment [a, b] n+1 points (i = 0, 1, 2, …, n) are given, named nodes of interpolation, and values of some functionin these points It is required to construct the interpolating function accepting in nodes of interpolation the same values, as, i.e. Vectorially it means (Fig. 1), that it is required to find some curve of the certain type which are passing through the given set of points.
Fig. 1. Geometrical representation of function interpolation
In such statement the problem of interpolation, generally speaking, can have either uncountable set of solutions or not have at all solutions. However the problem it becomes unequivocal solved if instead of any function to search a polynomial of degree not above n, satisfying to conditions:
Received interpolation formulais used for approached calculation of values of the given functionfor those х which are distinct from nodes of interpolation. Such operation refers to as interpolation of function.