Results 151 to 160 of about 31,275 (174)

Estimates of the Remainder in Taylor's Formula

Series on Concrete and Applicable Mathematics, 2010
exaly   +2 more sources

Mind the Remainder: Taylor’s Theorem View on Recurrent Neural Networks

IEEE Transactions on Neural Networks and Learning Systems, 2022
Recurrent neural networks (RNNs) have gained tremendous popularity in almost every sequence modeling task. Despite the effort, these kinds of discrete unstructured data, such as texts, audio, and videos, are still difficult to be embedded in the feature space.
Xiang Guan, Yang Yang 0002, Jingjing Li 0001, Xing Xu 0001, Heng Tao Shen   +4 more
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On the Lagrange Remainder of the Taylor Formula

The American Mathematical Monthly, 2003
(2003). On the Lagrange Remainder of the Taylor Formula. The American Mathematical Monthly: Vol. 110, No. 7, pp. 627-633.
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Fluctuationless Univariate Integration Through Taylor Expansion with Remainder by Using Oscillatory Function Basis Sets

, 2009
This work uses a recently developed fluctuation free matrix representation method in approximating the integral of the Taylor expansion remainder term. The basis set used for the matrix representation contains common factors of sine and cosine functions ...
Ercan Gürvit, Metin Demiralp, Baykara N A   +2 more
exaly   +3 more sources

Computation and Application of Taylor Polynomials with Interval Remainder Bounds

Reliable Computing, 1998
So-called Taylor models are used to determine guaranteed bounds of function values of multivalued and preferably complicated functions which are expansive to evaluate. A Taylor model of a function \(f\) consists of a Taylor polynomial of some convenient degree and an absolute error term in form of an interval. In order to determine the required bounds,
Martin Berz, Georg Hoffstätter
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Taylor-Like Remainder Formulas for Interpolation by Arbitrary Linear Functionals

SIAM Journal on Numerical Analysis, 1974
Let $\mathcal{L}^i (i = 1,2, \cdots ,n)$ denote n linear functionals on $C^n [a,b]$, $f \in C^n [a,b]$, $p \in P_{n - 1} $ (the space of polynomials of degree $ < n$), where $\mathcal{L}^i p = \mathcal{L}^i f$. Then one has the formula $| {f(x) - p(x)} | \leqq U(x)\| {f^{(n)} } \|_\infty $, where U is the upper envelope of all functions $R \in C^n [a,b]
Chalmers, Bruce L., Metcalf, Frederic T.
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Approximating Csiszár f-divergence by the use of Taylor's formula with integral remainder

Mathematical Inequalities & Applications, 2002
Csiszár \(f\)-divergence is defined by \[ D_f(p,q):= \int_\Gamma p(x)f\Biggl[{q(x)\over p(x)}\Biggr] d\mu(x),\quad p,q\in\Omega, \] where \(f\) is convex on \((0,\infty)\), a set \(\Gamma\) and the \(\sigma\)-finite measure \(\mu\) are given and \(\Omega\) is the set of all probability densities on \(\mu\); that is \[ \Omega:= \Biggl\{p\mid p:\Gamma\to
Barnett, N. S., Cerone, P., Dragomir, S. S., Sofo, A.   +3 more
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The Remainder Term of the Taylor Expansion for a Holomorphic Function Is Representable in Lagrange Form

Siberian Mathematical Journal, 2003
Summary: We show that the remainder of the Taylor expansion for a holomorphic function can be written down in Lagrange form, provided that the argument of the function is sufficiently close to the interpolation point. Moreover, the value of the derivative in the remainder can be taken in the intersection of the disk whose diameter joins the ...
Radzievskaya, E. I., Radzievskij, G. V.
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Multi Nodalset Fluctuation Free Integration in Taylor Remainder’s Evaluation

AIP Conference Proceedings, 2010
The matrix representation of a univariate function is equal to the image of the independent variable matrix representation under that function at the no fluctuation limit. In recent studies of BEBBYT group this fact is extended in such a way that the matrix representation of a univariate function can be expressed as a linear combination of the same ...
Ercan Gürvit, N. A. Baykara, Metin Demiralp, Theodore E. Simos, George Psihoyios, Ch. Tsitouras   +5 more
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Some integral inequalities involving Taylor's remainder. I

2002
By applying the famous Steffensen inequality, the author deduces some interesting bounds for Taylor's remainder. As particular cases he reobtains a known Iyengar integral inequality, as well as one of the Hermite-Hadamard inequalities for convex functions.
openaire   +2 more sources