Thiele's interpolation formula

In mathematics, Thiele's interpolation formula is a formula that defines a rational function from a finite set of inputs and their function values . The problem of generating a function whose graph passes through a given set of function values is called interpolation. This interpolation formula is named after the Danish mathematician Thorvald N. Thiele. It is expressed as a continued fraction, where ρ represents the reciprocal difference: Be careful that the -th level in Thiele's interpolation formula is while the -th reciprocal difference is defined to be .