Reciprocal difference

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Reciprocal difference

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

:$ho_1\left(x_0, x_1\right) = frac\left\{x_0 - x_1\right\}\left\{f\left(x_0\right) - f\left(x_1\right)\right\}$

:$ho_2\left(x_0, x_1, x_2\right) = frac\left\{x_0 - x_2\right\}\left\{ ho_1\left(x_0, x_1\right) - ho_1\left(x_1, x_2\right)\right\} + f\left(x_1\right)$

:$ho_n\left(x_0,x_1,ldots,x_n\right)=frac\left\{x_0-x_n\right\}\left\{ ho_\left\{n-1\right\}\left(x_0,x_1,ldots,x_\left\{n-1\right\}\right)- ho_\left\{n-1\right\}\left(x_1,x_2,ldots,x_n\right)\right\}+ ho_\left\{n-2\right\}\left(x_1,ldots,x_\left\{n-1\right\}\right)$

ee also

*divided differences

References

*MathWorld|title=Reciprocal Difference|urlname=ReciprocalDifference
*citebook|last=Abramowitz|first=Milton|coathor=Irene A. Stegun|title=Handbook of Mathematical Functions|origyear=1964|year=1972|publisher=Dover|edition=ninth Dover printing, tenth GPO printing|pages=878|id=ISBN 0-486-61272-4

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