Thiele's interpolation formula

In mathematics, Thiele's interpolation formula is a formula that defines a rational function [math]\displaystyle{ f(x) }[/math] from a finite set of inputs [math]\displaystyle{ x_i }[/math] and their function values [math]\displaystyle{ f(x_i) }[/math]. 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:

[math]\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}}} }[/math]

Note that the [math]\displaystyle{ n }[/math]-th level in Thiele's interpolation formula is

[math]\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}, }[/math]

while the [math]\displaystyle{ n }[/math]-th reciprocal difference is defined to be

[math]\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}) }[/math].

The two [math]\displaystyle{ \rho_{n-2} }[/math] terms are different and can not be cancelled!

References

• Weisstein, Eric W.. "Thiele's Interpolation Formula". http://mathworld.wolfram.com/ThielesInterpolationFormula.html.