Taylor series expansion, linear

The method developed for linear constraints is extended to nonlinearly constrained problems. It is based on the idea that the nonlinear constraints measurement values are used as initial estimations for the measured process variables. The following linear system of equations is obtained ... 

Improved estimates of the parameters can be obtained by a differential correction technique based on least squares, provided that the estimates are sufficiently close to the actual values of the parameters A to lead to convergence of the method. This differential correction technique can be derived by first expanding the function about a using a linear Taylor series expansion of the form... 

To continue, it is necessary to find a method to evaluate J or J at point 2. To do this, a linear Taylor series expansion can be used. The general form of a Taylor series expansion is shown in Eq. 4.14 ... 

For a linear Taylor series expansion, the HOTs are ignored. Substituting 4.15 and 4.16 into the net flux Eq. 4.13, and dropping the subscript 1, gives ... 

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

Lindstrom and Bates argue that a Taylor series expansion of (Eq. 3.4) around the expectation of the random effects bi = 0 may be poor. Instead, they consider linearizing (Eq. 3.4) in the random effects about some value bf closer to bi than its expectation 0. 

By using a Taylor series expansion on the right hand side of Equation 6.72 and keeping only the linear terms we obtain the following equation... 

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. 

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. 

We develop y into a Taylor-series around a set of initial values for the k- which must not deviate too much from the optimised final values. With this condition we may truncate the Taylor-expansion after the linear terms and obtain the following system of linear relations ... 

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... 

Assuming Taylor series expansion using only zero- and first-order terms (dropping second and higher order terms), we arrive at the linear or linearized system described by... 

To apply the procedure, the nonlinear constraints Taylor series expansion and an optimization problem is resolved to find the solution, d, that minimizes a quadratic objective function subject to linear constraints. The QP subproblem is formulated as follows ... 

The linear expansion of the natural logarithm in a Taylor series... 

Using once more the linear expansion of the natural logarithm in a Taylor series for the second term between brackets allows us to find an approximation valid for S[Pg.51]

In general, non-linear problems cannot be resolved explicitly, i.e. there is no equation that allows the computation of the result in a direct way. Usually such systems can be resolved numerically in an iterative process. In most instances, this is done via a truncated Taylor series expansion. This downgrades the problem to a linear one that can be resolved with a stroke of the brush or the Matlab / and commands see The Pseudo-Inverse (p.ll 7). 

Linearization is quite straightforward. All we do is take the nonlinear functions, expand them in Taylor series expansions around the steadystate operating level, and neglect all terms after the first partial derivatives. 

In a further simplification, namely the expansion of the exponential function in Eq. (2.13) into a Taylor series up to the linear term only (neglecting other terms which may be shown to be negligible) yields... 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

If V(r, x) were a known function, this linear expansion could be used to determine how the velocity varies for short intervals of time and in any arbitrary short spatial direction dx. In a Taylor-series expansion of a scalar field, it is often conventional to post-multiply by the dx. Since the gradient of a scalar field is a vector and because the inner product of two vectors is commutative, the order of the product is unimportant. However, because of the tensor structure of the gradient of a vector field, the pre-multiply is essential. 

The most prominent of these methods is probably the second order Newton-Raphson approach, where the energy is expanded as a Taylor series in the variational parameters. The expansion is truncated at second order, and updated values of the parameters are obtained by solving the Newton-Raphson linear equation system. This is the standard optimization method and most other methods can be treated as modifications of it. We shall therefore discuss the Newton-Raphson approach in more detail than the alternative methods. 

The solution procedure was as follows Linearize Eq. (7.2b) by assuming v = const = Q/rcR2. Here Q is volume productivity of the plasticizing unit which is constant during the entire filling process and determined by its plastication parameters. Then find a solution to Eq. (7.2b) by the well-known methods of mathematical physics, and substitute it into Eq. (7.1a). Linearization of Eq. (7.2a) is performed by the expansion of L /KT(r, t) into the Taylor series. From Eq. (7.2a) and condition Q = const, we find the expression for v(r) ... 

Having gone thus far with the OF-KEDF s, one ultimately faces the most difficult problem how to make the entire OF-DFT scheme, especially the evaluation of the DD AWF, linear-scaling with respect to the system size. This is a general numerical bottleneck of all the NLDA s, as discussed in Section V the presence of DD terms inside the AWF in Eq. (164) makes a straightforward application of the FFT impossible. However, one can use a Taylor series expansions to factor... 

Let us denote the numerical solutions 4>i, Taylor series expansion (Equation (A. 15)) will not change with grid size, the following linear system of equations should be solved... 

The Volterra Series. For a time invariant system defined by equation 4.25, it is possible to form a Taylor series expansion of the non-linear function to give [Priestley, 1988] ... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

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... 

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. 

Equation 7 is linearized about an initial point x using the Taylor series expansion. 

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... 

The accuracy of the averaged model truncated at order p9(q 0) thus depends on the truncation of the Taylor series as well as on the truncation of the perturbation expansion used in the local equation. The first error may be determined from the order pq 1 term in Eq. (23) and may be zero in many practical cases [e.g. linear or second-order kinetics, wall reaction case, or thermal and solutal dispersion problems in which / and rw(c) are linear in c] and the averaged equation may be closed exactly, i.e. higher order Frechet derivatives are zero and the Taylor expansion given by Eq. (23) terminates at some finite order (usually after the linear and quadratic terms in most applications). In such cases, the only error is the second error due to the perturbation expansion of the local equation. This error e for the local Eq. (20) truncated at 0(pq) may be expressed as... 

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