Volume 24, Issue 4
On the Divided Difference Form of Faà Di Bruno's Formula
DOI:

J. Comp. Math., 24 (2006), pp. 553-560

Published online: 2006-08

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• Abstract

The $n$-divided difference of the composite function $h:=f\circ g$ of functions $f$, $g$ at a group of nodes $t_0, t_1, \cdots, t_n$ is shown by the combinations of divided differences of $f$ at the group of nodes $g(t_0), g(t_1), \cdots, g(t_m)$ and divided differences of $g$ at several partial group of nodes $t_0, t_1,\cdots, t_n$, where $m=1, 2,\cdots, n$. Especially, when the given group of nodes are equal to each other completely, it will lead to Fa\{a} di Bruno's formula of higher derivatives of function $h$.

• Keywords

Divided difference Newton interpolation Composite function FAA di Bruno's formula Bell polynomial

@Article{JCM-24-553, author = {}, title = {On the Divided Difference Form of Faà Di Bruno's Formula}, journal = {Journal of Computational Mathematics}, year = {2006}, volume = {24}, number = {4}, pages = {553--560}, abstract = { The $n$-divided difference of the composite function $h:=f\circ g$ of functions $f$, $g$ at a group of nodes $t_0, t_1, \cdots, t_n$ is shown by the combinations of divided differences of $f$ at the group of nodes $g(t_0), g(t_1), \cdots, g(t_m)$ and divided differences of $g$ at several partial group of nodes $t_0, t_1,\cdots, t_n$, where $m=1, 2,\cdots, n$. Especially, when the given group of nodes are equal to each other completely, it will lead to Fa\{a} di Bruno's formula of higher derivatives of function $h$. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8774.html} }
TY - JOUR T1 - On the Divided Difference Form of Faà Di Bruno's Formula JO - Journal of Computational Mathematics VL - 4 SP - 553 EP - 560 PY - 2006 DA - 2006/08 SN - 24 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8774.html KW - Divided difference KW - Newton interpolation KW - Composite function KW - FAA di Bruno's formula KW - Bell polynomial AB - The $n$-divided difference of the composite function $h:=f\circ g$ of functions $f$, $g$ at a group of nodes $t_0, t_1, \cdots, t_n$ is shown by the combinations of divided differences of $f$ at the group of nodes $g(t_0), g(t_1), \cdots, g(t_m)$ and divided differences of $g$ at several partial group of nodes $t_0, t_1,\cdots, t_n$, where $m=1, 2,\cdots, n$. Especially, when the given group of nodes are equal to each other completely, it will lead to Fa\`{a} di Bruno's formula of higher derivatives of function $h$.