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:

:$f(x)\; =\; f(x\_1)\; +\; \backslash cfrac\{x-x\_1\}\{\backslash rho(x\_1,x\_2)\; +\; \backslash cfrac\{x-x\_2\}\{\backslash rho\_2(x\_1,x\_2,x\_3)\; -\; f(x\_1)\; +\; \backslash cfrac\{x-x\_3\}\{\backslash rho\_3(x\_1,x\_2,x\_3,x\_4)\; -\; \backslash rho(x\_1,x\_2)\; +\; \backslash cdots\}\}\}$

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

:$\backslash rho\_n(x\_1,x\_2,\backslash cdots,x\_\{n+1\})-\backslash rho\_\{n-2\}(x\_1,x\_2,\backslash cdots,x\_\{n-1\})+\backslash cfrac\{x-x\_\{n+1\}\}\{\backslash rho\_\{n+1\}(x\_1,x\_2,\backslash cdots,x\_\{n+2\})-\backslash rho\_\{n-1\}(x\_1,x\_2,\backslash cdots,x\_\{n\})+\backslash cdots\},$

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

:$\backslash rho\_n(x\_1,x\_2,\backslash ldots,x\_\{n+1\})=\backslash frac\{x\_1-x\_\{n+1\}\}\{\backslash rho\_\{n-1\}(x\_1,x\_2,\backslash ldots,x\_\{n\})-\backslash rho\_\{n-1\}(x\_2,x\_3,\backslash ldots,x\_\{n+1\})\}+\backslash rho\_\{n-2\}(x\_2,\backslash ldots,x\_\{n\})$.

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