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BIT Numerical Mathematics

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01-09-2012

Divided differences of multivariate implicit functions

Author: Georg Muntingh

Published in: BIT Numerical Mathematics | Issue 3/2012

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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.

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Title: Divided differences of multivariate implicit functions
Author: Georg Muntingh
Publication date: 01-09-2012
Publisher: Springer Netherlands
Published in: BIT Numerical Mathematics / Issue 3/2012
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-012-0377-1

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