First-order Taylor series expansion

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. 

Laplace transform is only applicable to linear systems. Hence, we have to linearize nonlinear equations before we can go on. The procedure of linearization is based on a first order Taylor series expansion. 

What we do is a freshmen calculus exercise in first order Taylor series expansion about the... 

In case you forgot, the first order Taylor series expansion can be written as... 

As soon as we finish the first-order Taylor series expansion, the equation is linearized. All steps that follow are to clean up the algebra with the understanding that terms of the steady state equation should cancel out, and to change the equation to deviation variables with zero initial condition. 

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... 

Net Forces on a Differential Control Volume Based on a differential control volume (i.e vanishingly small dimensions in each of three spatial coordinates), we write the forces on each of the six faces of the control volume. The forces are presumed to be smooth, continuous, differentiable, functions of the spatial coordinates. Therefore the spatial variations across the control volume in each coordinate direction may be represented as a first-order Taylor-series expansion. When the net force is determined on the differential control volume, each term will be the product a factor that is a function of the velocity field and a factor that is the volume of the differential control volume 8 V. 

Consider the two-dimensional stresses on the faces of a cartesian control volume as illustrated in Fig. 2.25. The differential control-volume dimensions are dx and dy, with the dz = 1. Assuming differential dimensions and that the stress state is continuous and differentiable, the spatial variation in the stress state can be expressed in terms of first-order Taylor series expansions. 

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]. 

The FOCE method uses a first-order Taylor series expansion around the conditional estimates of the t] values. This means that for each iteration step where population estimates are obtained the respective individual parameter estimates are obtained by the FOCE estimation method. Thus, this method involves minimizations within each minimization step. The interaction option available in FOCE considers the dependency of the residual variability on the interindividual variability. The Laplacian estimation method is similar to the FOCE estimation method but uses a second-order Taylor series expansion around the conditional estimates of the 77 values. This method is especially useful when a high degree of nonlinearity occurs in the model [10]. 

To avoid the complex form of the error function, simplified solutions have been proposed in the literature [10]. To solve for the ignition delay time (tP fig), a first-order Taylor series expansion of Equation 3.19 is conducted. The range of validity of this expansion is limited, and thus, cannot be used over a large range of incident heat fluxes. Therefore, the domain has to be divided at least into two. 

The first domain corresponds to high-incident heat fluxes, where the pyrolysis temperature (TP) is attained very fast, thus t Application of the first-order Taylor Series expansion to Equation 3.13 around tp/tc —> 0 yields the following formulation for the pyrolysis time (lp) ... 

Class II Methods. The methods of Class II are those that use the simultaneous Newton-Raphson approach, in which all the equations are linearized by a first order Taylor series expansion about some estimate of the primitive variables. In its most general form, this expansion includes terms arising from the dependence of the thermo-physical property models on the primitive variables. The resulting system of linear equations is solved for a set of iteration variable corrections, which are then applied to obtain a new estimate. This procedure is repeated until the magnitudes of the corrections are sufficiently small. 

A linear approximation to the function y for points x in the neighborhood of a is given by the first-order Taylor series expansion... 

First-Order (NONMEM) Method. The first nonlinear mixed-effects modeling program introduced for the analysis of large pharmacokinetic data was NONMEM, developed by Beal and Sheiner. In the NONMEM program, linearization of the model in the random effects is effected by using the first-order Taylor series expansion with respect to the random effect variables r], and Cy. This software is the only program in which this type of linearization is used. The jth measurement in the ith subject of the population can be obtained from a variant of Eq. (5) as follows ... 

The first-order Taylor series expansion of the above model with respect to the random variables t, and y around zero is given by... 

This approach is called the first order (FO) method in NONMEM. This is the most widely used approach in population pharmacokinetic and pharmacodynamic data analysis, and has been evaluated by simulation. The use of the first-order Taylor series expansion to approximate the non-linear model in r], and, possibly,... 

Conditional First-Order (NLME) Method. Proposed by Lindstrom and Bates,this uses a first-order Taylor series expansion about conditional estimates of interindividual random effects. This estimation method is available in S-plus statistical software as the function NLME. ... 

The first tenn represents the change in / due to changes in the independent variable x, and the second term representce changes due to the independent variable y. Note that Eq. (1.7) is just a generalization of a first order Taylor series expansion to a function of two variables. 

For completeness, we should mention that inclusion of single-phonon events can be accomplished within a time-independent scattering theory simply by using a first-order Taylor series expansion of the interaetion potential between the gas molecule and the solid surface ... 

Eq. 6.2.6 was solved analytically to obtain the operation curve of the reactor (X vs t). Lumped kinetic parameters were determined by non-linear regression of experimental data using the numerical method of Newton-Raphson with first-order Taylor series expansion. Lumped parameters were smooth functions of temperature all parameters were adequately fitted to second order polynomials except for D that required a fourth order polynomial. The model can be used for reactor temperature optimization and can be extended to prolonged sequential batch operation provided that a sound model for enzyme inactivation is validated (Illanes et al. 2005b). 

Probabilistic response analysis consists of computing the probabilistic characterization of the response of a specific structure, given as input the probabilistic characterization of material, geometric and loading parameters. An approximate method of probabilistic response analysis is the mean-centred First-Order Second-Moment (FOSM) method, in which mean values (first-order statistical moments), variances and covariances (second-order statistical moments) of the response quantities of interest are estimated by using a mean-centred, first-order Taylor series expansion of the response quantities in terms of the random/uncertain model parameters. Thus, this method requires only the knowledge of the first- and second-order statistical moments of the random parameters. It is noteworthy that often statistical information about the random parameters is limited to first and second moments and therefore probabilistic response analysis methods more advanced than FOSM analysis cannot be fully exploited. 

The slope or derivative of f(x) is computed at an initial guess, a, for the root of f x) = 0. The new value of the root, 6, is computed based on a first-order Taylor Series expansion of f x) about the initial guess, a,... 

The analysis was based on a linear relation between the flow and the force. Far from equilibrium this linear relation will not be accurate anymore. However extension to this range will be possible using a first order Taylor series expansion for the relation between the flow and the force. 

For small excursion of the runner about its equilibrium position, the non linear functions (5) can be linearized with adequate accuracy for most practical purposes. The force and moment can be represented by their first order Taylor series expansion e.g. 

Since the EKF is based on the first-order Taylor series expansion, the accuracy and stability of the EKF may not be sufficient for many applications with large uncertainties. Many quadrature-based Gaussian approximation filters can be used in the same filtering framework to improve the performance of the EKF. 

An important method for calculating approximate confidence intervals is through linearization with a first-order Taylor series expansion around the estimated parameter, which results in... 

Within the EKF both process and measurement models are linearized through a first-order Taylor series expansion around the current state estimate. Here the original nonlinear functions are used for the state transition and the measurement prediction, while the covariances are approximated by calculating the associated Jacobian matrices. 

As also mentioned before, the limit state function is generally nonUnear. Therefore, the function g(U) usually cannot be characterized by a first-order polynomial. An approximate way to solve this problem is to replace the nonlinear function with a first-order Taylor series expansion. In other words, the performance surface, in the neighborhood of the design point Up, is approximated by the tangent hyperplane at Up (Fig. 4b). In analytical terms, the approximate limit state function gf(U) becomes... 

In the two previous chapters the code segments developed, in particular new-ton() and nsolv() used a numerical derivative for the first-order Taylor series expansions of various functions. In these routines the simple single sided derivative equation was used with a relative default displacement factor of 10 . The single sided derivative was used because in all the routines involving Newton s method, the funetion value is required in addition to the derivative and thus the single sided derivative requires only one additional funetion evaluation whereas the use of the... 

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