Divided Differences of Multivariate Implicit Functions

Abstract

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 $g$.

BIT Numerical Mathematics (2012)

References in corpus (1)

• Divided Differences of Implicit Functions