A Second-Order Taylor Expansion

Luenberger. Optimization by Vector Space Methods, Chapter 7, pages 169-175. John Wiley Sons Inc., New York, 1969. 

Nashed. Some remarks on variations and differentials. Am. Math. Mon., 

Variational Methods in Optimization, Chapter 1, pages 9-30. Dover Publications Inc., New York, 1998. 

2 In the plug flow reactor problem of Section 1.3.2 (p. 6), substitute Elqua-tion (1.11) into Equation (1.12) and And the variation of the resulting objective functional. 

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A further improvement can be seen for the situation depicted in Eigure lb. Let ( )i, (r) denote the potential due to the charges in the cell about point b, evaluated at the point r. Let a be the center of the subcell containing q. Then (j), (r) can be approximated by a second-order Taylor expansion about a ... 

The potential energy is approximated by a second-order Taylor expansion around the stationary geometry. 

Certainly, nonlinearities in real data can have several possible causes, both chemical (e.g., interactions that make the true concentrations of any given species different than expected or might be calculated solely from what was introduced into a sample, and interaction can change the underlying absorbance bands, to boot) and physical (such as the stray light, that we simulated). Approximating these nonlinearities with a Taylor expansion is a risky procedure unless you know a priori what the error bound of the approximation is, but in any case it remains an approximation, not an exact solution. In the case of our simulated data, the nonlinearity was logarithmic, thus even a second-order Taylor expansion would be of limited accuracy. 

The step size control parameter R, initially set to a value of order unity, is adaptive, in the sense that it is decreased (or increased) at each iteration depending on how well (or how badly) the energy change actually brought about by the corrections accords with the value predicted from a second-order Taylor expansion in the corrections themselves. In extreme cases, the corrections are rejected and recomputed with an increased value of R. Otherwise, any updates to R apply from the next iteration. The precise set of rules used to control R may affect efficiency but is not critical to the success of the minimization procedure, as long as the rules provide the correct qualitative behaviour (e.g. see Refs. [22] and [18]). 

Let 9m denote the mode (initially unknown) of the log-likelihood function L 9 y) for the single parameter 9. A second-order Taylor expansion of L around 9m, for a given model and pattern of experiments, gives... 

When data are obtained, we assume that the log-likelihood function will have a local maximum (mode) at some point 6 = 6m- A second-order Taylor expansion of L(6) around that point will give... 

A semiempirical model that has been particularly useful in zeolite studies is known as the electronegativity equalization method (EEM). EEM is based on the following assumptions (1) the electron density can be partitioned into spherical atomic contributions (2) T[p] and Ve [p] in Eq. [6] can be written as a second-order Taylor expansion of the effective charges and (3) each atom i carries an effective charge q,- = Z, - N,, where Z, and N, are the charge on the nucleus and the number of electrons in the atom, respectively. Then the total energy is given ... 

The solution of these equations may be obtained by use of the iterative Newton method. The Newton method can be derived from a second order Taylor expansion, which for the given Lagrangian (L) gives ... 

By a near-quadratic surface, we mean a PES like that of Eq. (4.9) or (4.14), where the PES is taken to be a second-order Taylor expansion in directions orthogonal to the reaction path, although we will include cases where some higher-order terms are included. 

If one is interested solely in the estimation of a characteristic decoherence time, an alternative to the generation and the analysis of a large set of diverging trajectories has been proposed by Schwartz et al Based on a second-order Taylor expansion of the positions and momentums of the nuclei at / = 0, the real part of the decoherence function can be approximated as ... 

The above results can be reasoned as follows. Given the data, let the MPV of the modal parameters be denoted by the vector 0. Approximating the NLLF by a second-order Taylor expansion about 0,... 

Since, in general, the atomic charges are not an integer, a second-order Taylor expansion of the energy, around the neutral state, is usually used for the quasi-atomic energies ... 

For the N-atom system, the energy V at a geometry (denoted by x in Cartesian coordinates and by q in internal coordinates) close to a reference geometry (denoted by x° in Cartesian coordinates and by q° in internal coordinates) can be obtained by a second-order Taylor expansion. In unsealed Cartesian and curvilinear coordinates, the expansions are given by... 

When distributing this term over the fluid by means of the filter kernel, it provides the source term Sp >f to the macroscopic momentum balance of the fluid. An improved expansion would be a second-order Taylor expansion of the filter function where the extra contribution is... 

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