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:

${\displaystyle \rho _{1}(x_{1},x_{2})={\frac {x_{1}-x_{2}}{f(x_{1})-f(x_{2})}}}$

${\displaystyle \rho _{2}(x_{1},x_{2},x_{3})={\frac {x_{1}-x_{3}}{\rho _{1}(x_{1},x_{2})-\rho _{1}(x_{2},x_{3})}}+f(x_{2})}$

${\displaystyle \rho _{n}(x_{1},x_{2},\ldots ,x_{n+1})={\frac {x_{1}-x_{n+1}}{\rho _{n-1}(x_{1},x_{2},\ldots ,x_{n})-\rho _{n-1}(x_{2},x_{3},\ldots ,x_{n+1})}}+\rho _{n-2}(x_{2},\ldots ,x_{n})}$

See also

• Divided differences

References

• Weisstein, Eric W. "Reciprocal Difference". MathWorld .
• 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.