In mathematics, Thiele's interpolation formula is a formula that defines a rational function f ( x ) {\displaystyle f(x)} from a finite set of inputs x i {\displaystyle x_{i}} and their function values f ( x i ) {\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:
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 ) + ⋯ {\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 }}}}}}}
Note that the n {\displaystyle n} -th level in Thiele's interpolation formula is
ρ n ( x 1 , x 2 , ⋯ , x n + 1 ) − ρ n − 2 ( x 1 , x 2 , ⋯ , x n − 1 ) + x − x n + 1 ρ n + 1 ( x 1 , x 2 , ⋯ , x n + 2 ) − ρ n − 1 ( x 1 , x 2 , ⋯ , x n ) + ⋯ , {\displaystyle \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 {\displaystyle n} -th reciprocal difference is defined to be
ρ n ( x 1 , x 2 , … , x n + 1 ) = x 1 − x n + 1 ρ n − 1 ( x 1 , x 2 , … , x n ) − ρ n − 1 ( x 2 , x 3 , … , x n + 1 ) + ρ n − 2 ( x 2 , … , x n ) {\displaystyle \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 ρ n − 2 {\displaystyle \rho _{n-2}} terms are different and can not be cancelled.
References Weisstein, Eric W. "Thiele's Interpolation Formula". MathWorld.