Taylor series expansion approximations

Evidently, this fomuila is not exact if fand vdo not connnute. However for short times it is a good approximation, as can be verified by comparing temis in Taylor series expansions of the middle and right-hand expressions in (A3,11,125). This approximation is intrinsically unitary, which means that scattering infomiation obtained from this calculation automatically conserves flux. 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

The simple harmonie motion of a diatomie moleeule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is plaeed on polyatomie moleeules whose eleetronie energy s dependenee on the 3N Cartesian eoordinates of its N atoms ean be written (approximately) in terms of a Taylor series expansion about a stable loeal minimum. We therefore assume that the moleeule of interest exists in an eleetronie state for whieh the geometry being eonsidered is stable (i.e., not subjeet to spontaneous geometrieal distortion). 

Finally, we show how to relate the modified Schrodinger equation evolution X(m) to the usual evolution T (t) [14]. Consider the modified Schrodinger equation, Eq. (12). We approximate f H) in this equation with a first-order Taylor series expansion. 

This is a form that serves many purposes. The term in the denominator introduces a negative pole in the left-hand plane, and thus probable dynamic effects to the characteristic polynomial of a problem. The numerator introduces a positive zero in the right-hand plane, which is needed to make a problem to become unstable. (This point will become clear when we cover Chapter 7.) Finally, the approximation is more accurate than a first order Taylor series expansion.1... 

Taylor-series expansion, we can approximate /P(A ) in terms of the unsubstituted (pure hydride) reference value /P(A ), Eq. (3.71a), and successive corrections < l, (A ) for each substituent... 

Successive linear programming (SLP) methods solve a sequence of linear programming approximations to a nonlinear programming problem. Recall that if g,(x) is a nonlinear function and x° is the initial value for x, then the first two terms in the Taylor series expansion of gt(x) around x° are... 

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

S°, the system of differential equations specified in Eq. (5) can be approximated by a Taylor series expansion... 

The residuals r(p+8p) after the application of the shift vector, are approximated by a Taylor series expansion. With sufficient terms, any precision for the approximation can be achieved. 

The Taylor series expansion is always only an approximation and therefore the shift vector 8p will not result in the minimum directly. However, the new parameter vector p+8p will usually be better than the preceding p. Thus, an iterative process should move towards the optimal parameters. 

Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Refer to Fig. 12.1 and its caption for more detail. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. Any such property, P, can be expressed in terms of a Taylor series expansion over the displacements of the coordinates from their equilibrium positions,... 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

Functional Taylor series expansion of the functional minimized in Eq. (87), in powers of noK ") = [nGs( ) - gs( )] has been employed first, and Eq. (88) used in the last step. So E " is close to KS correlation energy functional taken for the GS density of HF approximation, corrected by the (much smaller) HF correlation energy, and a small remainder of the second order in the density difference. The last quantity gives an estimate to the large parentheses term of Eq. (28) in [12]. 

The rationale for using low degree polynomials to approximate / is based on a Taylor series expansion off around x=0. 

It is difficult to solve the system of Eqs. (39)—(41) for these boundary conditions. However, certain simplifying assumptions can be made, if the Prandtl number approaches large values. In this case, the thermal boundary layer becomes very thin and, therefore, only the fluid layer near the plate contributes significantly to the heat transfer resistance. The velocity components in Eq. (41) can then be approximated by the first term of their Taylor series expansions in terms of y. In addition, because the nonlinear inertial terms are negligible near the wall, one can further assume that the combined forced and free convection velocity is approximately equal to the sum of the velocities that would exist when these effects act independently. Therefore, for assisting flows at large Prandtl numbers (theoretically for Pr -> oo), Eq. (41) can be rewritten in the form ... 

Exercises Use the first few terms of the Taylor series expansion (1.23) to develop small-x approximations for the functions... 

Taylor-series expansions allow the development of finite differences on a more formal basis. In addition, they provide tools to analyze the order of the approximation and the error of the final solution. In order to introduce the methodology, let s use a simple example by trying to obtain a finite difference expression for dp/dx at a discrete point i, similar to those in eqns. (8.1) to (8.3). Initially, we are going to find an expression for this derivative using the values of

backward difference equation). Thus, we are looking for an expression such as... 

The function U = U(ft) can be approximated by the second-order Taylor series expansion at the point i.e.,... 

These equations are valid only for dilute solution because of approximations token in the Taylor-series expansion ... 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

For low-incident heat fluxes where tp L, the Taylor series expansion is made around tp/tc —> °°, where the first-order approximation yields... 

For more complex models or for input distributions for which exact analytical methods are not applicable, approximate methods might be appropriate. Many approximation methods are based on Taylor series expansion solutions, in which the series is truncated depending on the desired amount of solution accuracy and whether one wishes to consider covariance among the input distributions (Hahn Shapiro, 1967). These methods often go by names such as generation of system moments , statistical error propagation , delta method and first-order methods , as discussed by Cullen Frey (1999). 

Approximation methods can be useful, but as the degree of complexity of the input distributions or the model increases, in terms of more complex distribution shapes (as reflected by skewness and kurtosis) and non-linear model forms, one typically needs to carry more terms in the Taylor series expansion in order to produce an accurate estimate of percentiles of the distribution of the model output. Thus, such methods are often most widely used simply to quantify the mean and variance of the model output, although even for these statistics, substantial errors can accrue in some situations. Thus, the use of such methods requires careful consideration, as described elsewhere (e.g. Cullen Frey, 1999). 

Function (6.13) has a distinct maximum at g = g and p = p1, so the integration in (6.11) can be performed analytically by linearizing the classical dynamics around the trajectory (qt,Pt), since the linearization leads to an approximate Taylor series expansion in variables Sqo = qr0 — go and Spo = p o Po- The approximations used are the same as those used in... 

Using a Taylor series expansion in the TD it can be shown that, to a good approximation,78 the Voigt function can be written as V, a linear combination of Lorentzian, g (f) and Gaussian gad) functions having the same width, W = Wi = Wq, namely... 

At each cycle of the iterative process a new parameter shift vector, 5k, is calculated. To derive the formulae for the iterative refinement of k, we develop R as a function of k (starting from k = k0) into a Taylor series expansion. For sufficiently small 5k, the residuals, R(k + 5k), can be approximated by a Taylor series expansion. 

The Taylor series expansion is an approximation, and therefore the shift vector 5k is an approximation as well. However, the new parameter vector k + 5k will generally be better than the preceding k. Thus, an iterative process should always move toward the optimal rate constants. As the iterative fitting procedure progresses, the shifts, 5 k, and the residual sum of squares, ssq, usually decrease continuously. The relative change in ssq is often used as a convergence criterion. For example, the iterative procedure can be terminated when the relative change in ssq falls below a preset value ji, typically ji = 10 4. 

---