Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Divided differences of implicit functions

Authors: Georg Muntingh and Michael Floater
Journal: Math. Comp. 80 (2011), 2185-2195
MSC (2010): Primary 26A24; Secondary 05A17, 41A05, 65D05
Published electronically: April 12, 2011
MathSciNet review: 2813354
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Under general conditions, the equation $ g(x,y) = 0$ implicitly defines $ y$ locally as a function of $ x$. In this article, we express divided differences of $ y$ in terms of bivariate divided differences of $ g$, generalizing a recent result on divided differences of inverse functions.

References [Enhancements On Off] (What's this?)

  • [1] Carl de Boor, Divided differences, Surv. Approx. Theory 1 (2005), 46-69 (electronic). MR 2221566 (2006k:41001)
  • [2] Michael S. Floater and Tom Lyche, Two chain rules for divided differences and Faà di Bruno's formula, Math. Comp. 76 (2007), no. 258, 867-877. MR 2291840 (2008e:65023)
  • [3] Michael S. Floater and Tom Lyche, Divided differences of inverse functions and partitions of a convex polygon, Math. Comp. 77 (2008), no. 264, 2295-2308. MR 2429886 (2009e:05027)
  • [4] Michael S. Floater and Tom Lyche, A Chain Rule for Multivariate Divided Differences (2009), available at
  • [5] Xinghua Wang and Aimin Xu, On the divided difference form of Faà di Bruno's formula. II, J. Comput. Math. 25 (2007), no. 6, 697-704. MR 2359959 (2008h:65009)
  • [6] Cavaliere Francesco Faà di Bruno, Note sur une nouvelle formule de calcul différentiel, Quarterly J. Pure Appl. Math. 1 (1857), 359-360.
  • [7] Warren P. Johnson, The curious history of Faà di Bruno's formula, Amer. Math. Monthly 109 (2002), no. 3, 217-234. MR 1903577 (2003d:01019)
  • [8] G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc. 348 (1996), no. 2, 503-520. MR 1325915 (96g:05008)
  • [9] Tom Wilde, Implicit higher derivatives, and a formula of Comtet and Fiolet (2008-05-17), available at
  • [10] Louis Comtet and Michel Fiolet, Sur les dérivées successives d'une fonction implicite, C. R. Acad. Sci. Paris Sér. A 278 (1974), 249-251 (French). MR 0348055 (50:553)
  • [11] Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128 (57:124)
  • [12] Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin. MR 1676282 (2000k:05026)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 26A24, 05A17, 41A05, 65D05

Retrieve articles in all journals with MSC (2010): 26A24, 05A17, 41A05, 65D05

Additional Information

Georg Muntingh
Affiliation: CMA/Matematisk Institutt, P.B 1053, Blindern, N-0316, Oslo, Norway

Michael Floater
Affiliation: CMA/Matematisk Institutt, P.B 1053, Blindern, N-0316, Oslo, Norway

Received by editor(s): January 15, 2010
Received by editor(s) in revised form: September 24, 2010
Published electronically: April 12, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society