Reciprocal Difference

The reciprocal differences are closely related to the Divided Difference. The first few are explicitly given by

$$\rho(x_0, x_1)={x_0-x_1\over f_0-f_1}$$ (1)

$$\rho_2(x_0, x_1, x_2)={x_0-x_2\over\rho(x_0, x_1)-\rho(x_1, x_2)}+f_1$$ (2)

$$\rho_3(x_0, x_1, x_2, x_3)={x_0-x_3\over \rho_2(x_0, x_1, x_2)-\rho_2(x_1, x_2, x_3)}+\rho(x_1, x_2)$$ (3)

$$\rho_n(x_0, x_1, \ldots, x_n)={x_0-x_n\over\rho_{n-1}(x_0, \... ...\rho_{n-1}(x_1, \dots, x_n)}+\rho_{n-2}(x_1, \ldots, x_{n-1}).$$ (4)

See also Backward Difference, Central Difference, Divided Difference, Finite Difference, Forward Difference

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 878, 1972.

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 443, 1987.