In mathematics, Thiele's interpolation formula is a formula that defines a rational function ${\displaystyle f(x)}$ from a finite set of inputs ${\displaystyle x_{i}}$ and their function values ${\displaystyle f(x_{i})}$. 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:

${\displaystyle f(x)=f(x_{1})+{\cfrac {x-x_{1}}{\rho (x_{1},x_{2})+{\cfrac {x-x_{2}}{\rho _{2}(x_{1},x_{2},x_{3})-f(x_{1})+{\cfrac {x-x_{3}}{\rho _{3}(x_{1},x_{2},x_{3},x_{4})-\rho (x_{1},x_{2})+\cdots }}}}}}}$