Reciprocal difference

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

${\displaystyle {\rho }_1(x_1,x_2)=\frac{x_1-x_2}{f(x_1)-f(x_2)}}$

${\displaystyle {\rho }_2(x_1,x_2,x_3)=\frac{x_1-x_3}{{\rho }_1(x_1,x_2)-{\rho }_1(x_2,x_3)}+f(x_2)}$

${\displaystyle {\rho }_n(x_1,x_2,\dots ,x_{n+1})=\frac{x_1-x_{n+1}}{{\rho }_{n-1}(x_1,x_2,\dots ,x_n)-{\rho }_{n-1}(x_2,x_3,\dots ,x_{n+1})}+{\rho }_{n-2}(x_2,\dots ,x_n)}$

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. 

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