In mathematics, Thiele's interpolation formula is a formula that defines a rational function f ( x ) from a finite set of inputs x i and their function values f ( x i ) . 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:
f ( x ) = f ( x 1 ) + x − x 1 ρ ( x 1 , x 2 ) + x − x 2 ρ 2 ( x 1 , x 2 , x 3 ) − f ( x 1 ) + x − x 3 ρ 3 ( x 1 , x 2 , x 3 , x 4 ) − ρ ( x 1 , x 2 ) + ⋯ 