Two chain rules for divided differences and Faà di Bruno’s formula

Floater, Michael; Lyche, Tom

doi:10.1090/S0025-5718-06-01916-8

Two chain rules for divided differences and Faà di Bruno’s formula
HTML articles powered by AMS MathViewer

by Michael S. Floater and Tom Lyche;

Math. Comp. 76 (2007), 867-877

DOI: https://doi.org/10.1090/S0025-5718-06-01916-8

Published electronically: October 30, 2006

PDF | Request permission

Abstract:

In this paper we derive two formulas for divided differences of a function of a function. Both formulas lead to other divided difference formulas, such as reciprocal and quotient rules. The two formulas can also be used to derive Faà di Bruno’s formula and other formulas for higher derivatives of composite functions. We also derive a divided difference version of Faà di Bruno’s determinant formula.

References

Carl de Boor, Divided differences, Surv. Approx. Theory 1 (2005), 46–69. MR 2221566
Nira Dyn and Charles A. Micchelli, Piecewise polynomial spaces and geometric continuity of curves, Numer. Math. 54 (1988), no. 3, 319–337. MR 971706, DOI 10.1007/BF01396765
C. F. Faà di Bruno, Note sur une nouvelle formule de calcul differentiel, Quarterly J. Pure Appl. Math 1 (1857), 359–360.
M. S. Floater, Arc length estimation and the convergence of parametric polynomial interpolation, BIT 45 (2005), 679–694.
John A. Gregory, Geometric continuity, Mathematical methods in computer aided geometric design (Oslo, 1988) Academic Press, Boston, MA, 1989, pp. 353–371. MR 1022718
T. N. T. Goodman, Properties of $\beta$-splines, J. Approx. Theory 44 (1985), no. 2, 132–153. MR 794596, DOI 10.1016/0021-9045(85)90076-0
Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 201039
Warren P. Johnson, The curious history of Faà di Bruno’s formula, Amer. Math. Monthly 109 (2002), no. 3, 217–234. MR 1903577, DOI 10.2307/2695352
Charles Jordan, Calculus of finite differences, 3rd ed., Chelsea Publishing Co., New York, 1965. Introduction by Harry C. Carver. MR 183987
Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. MR 378456
Knut Mørken and Karl Scherer, A general framework for high-accuracy parametric interpolation, Math. Comp. 66 (1997), no. 217, 237–260. MR 1372007, DOI 10.1090/S0025-5718-97-00796-5
Elena Popoviciu, Sur quelques propriétés des fonctions quasi-convexes, Itinerant seminar on functional equations, approximation and convexity (Cluj-Napoca, 1983) Preprint, vol. 83, Univ. “Babeş-Bolyai”, Cluj-Napoca, 1983, pp. 107–114 (French). MR 750504
Tiberiu Popoviciu, Introduction à la théorie des différences divisées, Bull. Math. Soc. Roumaine Sci. 42 (1940), no. 1, 65–78 (French). MR 13171, DOI 10.1016/0016-0032(57)90875-x
John Riordan, Derivatives of composite functions, Bull. Amer. Math. Soc. 52 (1946), 664–667. MR 17784, DOI 10.1090/S0002-9904-1946-08621-8
John Riordan, An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 96594
Steven Roman, The formula of Faà di Bruno, Amer. Math. Monthly 87 (1980), no. 10, 805–809. MR 602839, DOI 10.2307/2320788
J. F. Steffensen, Interpolation, Baltimore, 1927.
J. F. Steffensen, Note on divided differences, Danske Vid. Selsk. Math.-Fys. Medd 17 (1939), 1–12.