Thiele's interpolation formula explained

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)} + … }}}

Note that the

n -th level in Thiele's interpolation formula is\rho _{n(x} _{1,x} _{2, … ,x} _{n+1} ) - \rho _{n-2} (x _{1,x} _{2, … ,x} _{n-1} ) + \cfrac{x - x _{n+1} },

while the

n -th reciprocal difference is defined to be\rho _{n(x} _{1,x} _{2,\ldots,x} _{n+1} ) = (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