Reciprocal difference

In mathematics, the reciprocal difference of a finite sequence of numbers $(x_0, x_1, ..., x_n)$ on a function $f(x)$ is defined inductively by the following formulas:

$\rho_1(x_0, x_1) = \frac{x_0 - x_1}{f(x_0) - f(x_1)}$

$\rho_2(x_0, x_1, x_2) = \frac{x_0 - x_2}{\rho_1(x_0, x_1) - \rho_1(x_1, x_2)} + f(x_1)$

$\rho_n(x_0,x_1,\ldots,x_n)=\frac{x_0-x_n}{\rho_{n-1}(x_0,x_1,\ldots,x_{n-1})-\rho_{n-1}(x_1,x_2,\ldots,x_n)}+\rho_{n-2}(x_1,\ldots,x_{n-1})$

See also

• divided differences

References

• Weisstein, Eric W., "Reciprocal Difference", MathWorld.
• 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.