19. Taylor's Theorem

GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Taylor's Theorem

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Taylor's Theorem

Theorem. (Taylor) Every functional $J:\ in ${\ has the representation


for some $\ between $0$ and $1$, where $J^\ is the ${\ gradientvector evaluated at ${\, and $J^{\ is the ${\Hessian matrix of $J$ at ${\, i.e.,
$\ $\ $\ (52)
$\ $\ $\ (53)

Proof. See Goldstein [Goldstein 1967] p. 119. The Taylor infinite series is treated in Williamson and Crowell [Williamson et al. 1972]. The present form is typically more useful for computing bounds on the error incurred by neglecting higher order terms in the Taylor expansion.

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