First-order Taylor series

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... 

As discussed in Section (8.2), Equations (8.64) and (8.65) is a set of (n + m) nonlinear equations in the n unknowns x and tn unknown multipliers A.. Assume we have some initial guess at a solution (x,A). To solve Equations (8.64)-(8.65) by Newton s method, we replace each equation by its first-order Taylor series approximation about (x,A). The linearization of (8.64) with respect to x and A (the arguments are suppressed)... 

Given an initial guess x0 for x, Newton s method is used to solve Equation (8.84) for x by replacing the left-hand sidex>f (8.84) by its first-order Taylor series approximation at x0 ... 

The normalized reaction rate expressions are first linearized about the steady-state operating conditions (ss) using a first-order Taylor series,... 

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. 

Consider the net mass flow through the cylindrical differential element illustrated in Fig. 3.6. The following analysis makes no explicit reference to the scalar product of the flux vector and the outward normal, j ndA. Rather, it is based on a more direct observation of how mass diffuses into and out of the control volume. It is presumed that the spatial components of j are resolved into spatial components that are normal to the control-volume faces, jk,z, jk,r, and jk,e Further it is presumed that a positive value for a spatial component of jk means that the corresponding flux is in the direction of the positive coordinate. The components of the diffusive mass flux are presumed to be continuous and differentiable throughout the fluid. Therefore the flux components can be expanded in a first-order Taylor series to express the local variations in the flux. The net mass of species k that crosses the control surfaces diffusively is given by the incoming minus the outgoing mass transport. Consider, for example, transport in the radial direction ... 

The pressure is presumed to vary smoothly throughout the length of the channel, so it can be expanded in a first-order Taylor series. The wall shear stress is presumed to come from an empirical correlation, since, by assumption, the model does not consider radial variations in the axial velocity. The control-surface integrals can be evaluated simply to yield an equation for the axial momentum balance... 

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) ... 

The resulting C + E equations are nonlinear in unknowns nj, nj, and tt but In nj are iteration variables since nj occur in logarithmic terms. These equations are linearized using first-order Taylor Series (Newton-Raphson method), in the variables An , A (In nj), and ir, and with n nj are reduced to S + 1 + E linear equations in unknowns AN, A (In N), and tt. When extended to include P mixed phases, we nave shown that they are nearly identical to the equations of the RAND Method and have the same coefficient matrix. 

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. ... 

Refer to equations (L.5). For a single equation (and variable),/(jc) = 0, Newton s method uses the expansion of / ( c) in a first-order Taylor series about a reference point (a starting guess for the solution) jco. 

To apply Newton s method, expand each equation as a first-order Taylor series to get a set of linear equations at the point (jcio, X20). 

Equation 6.1 includes mass accumulation in a storage term in the differential volume, the mass transport by in- and outgoing convection and dispersion as well as the mass transfer into the particles. Using a first-order Taylor series approximation for the outgoing streams,... 

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 ... 

Another method for finding the minimum of a function, one that requires the function to be twice differentiable and that its derivatives can be calculated, is the Newton-Raphson algorithm. The algorithm begins with an initial estimate, xi, of the minimum, x. The goal is to find the value of x where the first derivative equals zero and the second derivative is a positive value. Taking a first-order Taylor series approximation to the first derivative (dY/dx) evaluated at xi... 

Further, let V denote the gradient and V2 denote the Hessian of the objective function evaluated at the current parameter estimates 0 . A first-order Taylor series approximation to S(0) about 0 can be written as... 

If g(0), be it univariate or multivariate, is a nonlinear function then an approach repeatedly seen throughout this book will be used—the function will first be linearized using a first-order Taylor series and then the expected value and variance will be found using Eqs. (3.55) and (3.56), respectively. This is the so-called delta method. If g(0) is a univariate, nonlinear function then to a first-order Taylor series approximation about 0 would be... 

If g(0) is a nonlinear function of two or more model parameters then the multivariate delta method can be used. For a function of two variables a first-order Taylor series approximation around 0 and 0j can be written as... 

Suppose Y = f(x, 0, t ) + g(z, e) where nr] — (0, il), (0, ), x is the set of subject-specific covariates x, z, O is the variance-covariance matrix for the random effects in the model (t ), and X is the residual variance matrix. NONMEM (version 5 and higher) offers two general approaches towards parameter estimation with nonlinear mixed effects models first-order approximation (FO) and first-order conditional estimation (FOCE), with FOCE being more accurate and computationally difficult than FO. First-order (FO) approximation, which was the first algorithm derived to estimate parameters in a nonlinear mixed effects models, was originally developed by Sheiner and Beal (1980 1981 1983). FO-approximation expands the nonlinear mixed effects model as a first-order Taylor series approximation about t) = 0 and then estimates the model parameters based on the linear approximation to the nonlinear model. Consider the model... 

Sheiner and Beal (1980 1981 1983) proposed taking a first-order Taylor series approximation around the set of r S evaluated at r = 0 to find the variance. Recall that Taylor series approximations, which are linear polynomials, take a function and create an approximation to the model around some neighborhood. The derivatives of Eq. (7.86) to the model are... 

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