Thiele's interpolation formula

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})+{\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 }}}}}}

References

Weisstein, Eric W. "Thiele's Interpolation Formula". MathWorld.

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