Reciprocal difference

In mathematics, the reciprocal difference of a finite sequence of numbers [math]\displaystyle{ (x_0, x_1, ..., x_n) }[/math] on a function [math]\displaystyle{ f(x) }[/math] is defined inductively by the following formulas:

[math]\displaystyle{ \rho_1(x_1, x_2) = \frac{x_1 - x_2}{f(x_1) - f(x_2)} }[/math]

[math]\displaystyle{ \rho_2(x_1, x_2, x_3) = \frac{x_1 - x_3}{\rho_1(x_1, x_2) - \rho_1(x_2, x_3)} + f(x_2) }[/math]

[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]

See also

• Divided differences

References

• Weisstein, Eric W.. "Reciprocal Difference". http://mathworld.wolfram.com/ReciprocalDifference.html. 
• Abramowitz, Milton; Irene A. Stegun (1972) [1964]. Handbook of Mathematical Functions (ninth Dover printing, tenth GPO printing ed.). Dover. p. 878. ISBN 0-486-61272-4.