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:

:$ho\_1(x\_0,\; x\_1)\; =\; frac\{x\_0\; -\; x\_1\}\{f(x\_0)\; -\; f(x\_1)\}$

:$ho\_2(x\_0,\; x\_1,\; x\_2)\; =\; frac\{x\_0\; -\; x\_2\}\{\; ho\_1(x\_0,\; x\_1)\; -\; ho\_1(x\_1,\; x\_2)\}\; +\; f(x\_1)$

:$ho\_n(x\_0,x\_1,ldots,x\_n)=frac\{x\_0-x\_n\}\{\; ho\_\{n-1\}(x\_0,x\_1,ldots,x\_\{n-1\})-\; ho\_\{n-1\}(x\_1,x\_2,ldots,x\_n)\}+\; ho\_\{n-2\}(x\_1,ldots,x\_\{n-1\})$

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