Abstract
This chapter presents some elementary properties of rational approximations and discusses the interpolation problem, the derivation of a rational function which takes specified values at a given set of points. The singularities of a rational function may enable the approximation to a function that is singular. There are well-known methods of systematic polynomial interpolation using finite differences or, more generally, divided differences. The basic formula is Newton's divided-difference interpolation formula. One useful property of the divided difference is that it is a symmetric function of its argument, unaffected by any permutation of the order of points in the original table. The same is not true of inverted differences. The construction of an inverted difference or reciprocal difference table involves a number of divisions, and it clearly fails if any divisor is zero.
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