Reciprocal difference

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title: "Reciprocal difference" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["finite-differences"] topic_path: "general/finite-differences" source: "https://en.wikipedia.org/wiki/Reciprocal_difference" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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:

:\rho_1(x_1, x_2) = \frac{x_1 - x_2}{f(x_1) - f(x_2)}

:\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)

:\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})

References

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finite-differences