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

:<math>\rho_1(x_1, x_2) = \frac{x_1 - x_2}{f(x_1) - f(x_2)}</math>

:<math>\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>\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==

*{{MathWorld|title=Reciprocal Difference|urlname=ReciprocalDifference}}
*{{cite book|last=Abramowitz|first=Milton|author2=Irene A. Stegun|title=[[Abramowitz and Stegun|Handbook of Mathematical Functions]]|orig-date=1964|year=1972|publisher=Dover|edition=ninth Dover printing, tenth GPO printing|isbn=0-486-61272-4|page=[https://archive.org/details/handbookofmathe000abra/page/878 878]}}

[[Category:Finite differences]]

Reciprocal difference - Revision history

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