Taylor Series Approximations

Alternative methods, such as correcting the nonlinearity though the application of an appropriate physical theory as we described above, may do as well or even better than a Taylor series approximation, but a rigorous theory is not always available. Even in... 

The error in this linear approximation approaches zero proportionally to (Ax)2 as Ax approaches zero. Given initial values for the variables, all nonlinear functions in the problem are linearized and replaced by their linear Taylor series approximations at this initial point. The variables in the resulting LP are the Ax/s, representing changes from the base vaiues. In addition, upper and lower bounds (called step bounds) are imposed on these change variables because the linear approximation is reasonably accurate only in some neighborhood of the initial point. 

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 ... 

Each property depends on only one (rather than the expected two) degrees of freedom, and each becomes pathological (divergent) in the limit of small T or P, respectively. For solids and liquids, aP and f3T are rather insensitive to P, T variations, so low-order Taylor series approximations may be adequate. For gases, however, it is generally necessary to differentiate an accurate equation of state to obtain a realistic (P, T) dependence of aP, fiT. 

The x - extends over all the solute species and is designated as xB, which we have previously used to designate the mole fraction of the single solute in a binary solution. At a concentration small enough to make the ideally dilute solution approximation, it is usually sufficient to use only the first term in the Taylor series approximation of ln(l — xB),4... 

This appendix reports the weights for the moments of the particle size distribution obtained from an eight order Taylor Series approximation to the scattering efficiency for the anomalous diffraction case... 

In many practical situations we have to compute a function / (A) of an x TV matrix A. A popular way of computing a matrix function is through the truncated Taylor series approximation. The conditions under which a matrix function / (A) has a Taylor series representation are given by the following theorem (Golub and Van Loan, 1996). 

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 result of a Taylor series approximation is that the computed shifts, Ap is not accurate yielding a fully minimized solution to the problem, but are a (hopefully) better approximation. Consequently, the new parameter values are used for a subsequent refinement cycle this process is repeated until the parameter shifts are less than some fraction of their estimated errors as obtained from the diagonal elements of the B matrix. 

The symbol 0() is known as the order symbol. It indicates the magnitude of the error in truncating the Taylor series approximation after only two terms, which is negligible in this case since 5x 2 < C Sx. It follows from (2-70) that the material point Q moves relative to the material point P with a velocity... 

To avoid this, we use domain perturbation theory (see Section E) to transform from the exact boundary conditions applied at rs = R + sf to asymptotically equivalent boundary conditions applied at the spherical surface rs = R(t). For example, instead of a condition on ur at r = R(t) + ef from the kinematic condition, we can obtain an asymptotically equivalent condition at r = R by means of the Taylor series approximation... 

Then, we relate the values of the various dependent variables such as Wq0) at z = h = 1 +h 8 + 0(52) to their value atz = 1 by using a simple Taylor series approximation, eg.,... 

A more direct motivation for studying linear flows is that we are frequently interested in applications of creeping-flow results for particles that are very small compared with the length scale L that is characteristic of changes in the undisturbed velocity gradient for a general flow. In this case, we may approximate the undisturbed velocity field in the vicinity of the particle by means of a Taylor series approximation, namely,... 

In the limit Pe p> 1, we have seen that the thermal boundary layer equations hold for a region very close to the surface of the body, and thus that we require only the limiting form of the velocity components as we approach the body surface. We can again use a Taylor series approximation to deduce the appropriate form for the velocity components,... 

The consequences of this change can be explored first in rather general terms without the need for reference to a specific problem. To see the general situation, it is sufficient to think in terms of a local 2D Cartesian coordinate system. The resulting analysis will be apphed directly only to a 2D problem. However, as we have seen in the preceding sections, the same qualitative result will be obtained for axisymmetric or even fully 3D problems. In the simplest view, the only difference between transport across a fluid interface and previous problems is in the Taylor series approximations for the velocity components (u, v). For convenience we assume that the local coordinate system is defined so the interface corresponds to y = 0. Because the first nonzero term for the tangential component is the shp velocity, the Taylor series approximation then takes the form,... 

Because the shape function h is unknown, we transform the boundary conditions to the undisturbed interface position at z7 = 0 by using the domain perturbation technique, which was introduced in previous chapters. Hence we can express eu[ at z = eh in terms of its value at z = 0 by using a Taylor series approximation ... 

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... 

A more efficient strategy would be to control both direction and distance. Consider a second-order Taylor series approximation to S(0) expanded around 0 ... 

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... 

Approximate (1 — a)100% confidence intervals can be developed using any of the methods presented in the bootstrapping section of the book appendix. Using the previous example, CL was simulated 1,000 times from a normal distribution with mean 50 L/h and variance 55 (L/h)2 while V was simulated 10,000 times with a mean of 150 L and variance 225 L2. The correlation between V and CL was fixed at 0.18 given the covariance matrix in Eq. (3.70). The simulated mean and variance of CL was 49.9 L/h and 55.5 (L/h)2, while the simulated mean and variance of V was 149.8 L with variance 227 L2. The simulated correlation between CL and V was 0.174. The mean estimated half life was 2.12 h with a variance of 0.137 h2, which was very close to the Taylor series approximation to the variance. The Sha-piro Wilk test for normality indicated that the distribution of half life was not normally distributed (p < 0.01). Hence, even though CL and V were normally distributed the resulting distribution for half life was not. Based on the 5 and 95% percentiles of the simulated half life... 

It is important to remember that for nonlinear functions the function is first linearized using a Taylor series approximation and then the mean and variance are calculated based on the approximation. How good the approximation is depends on how nonlinear g is around 0 and on the size of the variance of 0. Better approximations to g(0) can be found through higher order approximations. For example, a second order Taylor series about 0 leads to... 

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... 

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