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:

$f(x)=f(x_1)+\frac{x-x_1}{\rho (x_1,x_2)+\frac{x-x_2}{{\rho }_2(x_1,x_2,x_3)-f(x_1)+\frac{x-x_3}{{\rho }_3(x_1,x_2,x_3,x_4)-\rho (x_1,x_2)+\cdots }}}$

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

${\rho }_n(x_1,x_2,\cdots ,x_{n+1})-{\rho }_{n-2}(x_1,x_2,\cdots ,x_{n-1})+\frac{x-x_{n+1}}{{\rho }_{n+1}(x_1,x_2,\cdots ,x_{n+2})-{\rho }_{n-1}(x_1,x_2,\cdots ,x_n)+\cdots },$

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

${\rho }_n(x_1,x_2,\dots ,x_{n+1})=\frac{x_1-x_{n+1}}{{\rho }_{n-1}(x_1,x_2,\dots ,x_n)-{\rho }_{n-1}(x_2,x_3,\dots ,x_{n+1})}+{\rho }_{n-2}(x_2,\dots ,x_n)$.

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

References

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