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:

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

Note that the $n$-th level in Thiele's interpolation formula is

$\rho _{n}(x_{1},x_{2},\cdots ,x_{n+1})-\rho _{n-2}(x_{1},x_{2},\cdots ,x_{n-1})+{\cfrac {x-x_{n+1}}{\rho _{n+1}(x_{1},x_{2},\cdots ,x_{n+2})-\rho _{n-1}(x_{1},x_{2},\cdots ,x_{n})+\cdots }},$

while the $n$-th reciprocal difference is defined to be

$\rho _{n}(x_{1},x_{2},\ldots ,x_{n+1})={\frac {x_{1}-x_{n+1}}{\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})-\rho _{n-1}(x_{2},x_{3},\ldots ,x_{n+1})}}+\rho _{n-2}(x_{2},\ldots ,x_{n})$.

The two $\rho _{n-2}$ terms are different and can not be cancelled!

References

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