Divided Differences of Multivariate Implicit Functions

Details Divided Differences of Multivariate Implicit Functions Divided Differences of Multivariate Implicit Functions

2012-02-29

arxiv.org/abs/1202.6537v1

Mathematics

Numerical Analysis

BIT Numerical Mathematics (2012)

Scientific paper

10.1007/s10543-012-0377-1

Under general conditions, the equation $g(x^1, ..., x^q, y) = 0$ implicitly defines $y$ locally as a function of $x^1, ..., x^q$. In this article, we express divided differences of $y$ in terms of divided differences of $g$, generalizing a recent formula for the case where $y$ is univariate. The formula involves a sum over a combinatorial structure whose elements can be viewed either as polygonal partitions or as plane trees. Through this connection we prove as a corollary a formula for derivatives of $y$ in terms of derivatives of $g$.

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