Taylor series expansion, computational

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). 

The advantages of the Kumar equation of state are purely computational. The resulting expressions are approximations to the Panayiotou-Vera equation of state that will reduce to the proper forms for random conditions. Kumar et al. (1987) state that the expressions in Panayiotou and Vera (1982) differ because of errors in the Panayiotou and Vera work. The Vera and Panayiotou expressions have been shown to be correct with the methods described by High (Chapter 5, 1990). Thus, the discrepancies between the Kumar equation of state and the Panayiotou and Vera equation of state must occur in the approximations due to the Taylor series expansion. 

Four frequently used conventions exist for the definition of non-linear optical polarizabilities, leading to confusion in the realm of NLO. This has been largely clarified by Willets et al. (1992) and in their nomenclature we have used the Taylor series expansion (T convention), originally introduced by Buckingham (1967), where the factorials n are explicitly written in the expansion. Here the polarizabilities of one order all extrapolate to the same value for the static limit w— 0. /3 values in the second convention, the perturbation series (B), have to be multiplied by a factor of 2 to be converted into T values. This is the convention used most in computations following the sum-over-states method (see p. 136). The third convention (B ) is used by some authors in EFISHG experiments and is converted into the T convention by multiplication by a factor of 6. The fourth phenomenological convention (X) is converted to the T convention by multiplication by a factor of 4. 

The electrode length, 2L, is much smaller than the computational domain used in the fluid simulations. A separate mesh was therefore used for the mass-transfer calculations. Velocity components in the mass-transfer calculations were approximated hy their first terms in a Taylor series expansion in distance y from the wall ... 

This section deals with the construction of optimal higher order FDTD schemes with adjustable dispersion error. Rather than implementing the ordinary approaches, based on Taylor series expansion, the modified finite-difference operators are designed via alternative procedures that enhance the wideband capabilities of the resulting numerical techniques. First, an algorithm founded on the separate optimization of spatial and temporal derivatives is developed. Additionally, a second method is derived that reliably reflects artificial lattice inaccuracies via the necessary algebraic expressions. Utilizing the same kind of differential operators as the typical fourth-order scheme, both approaches retain their reasonable computational complexity and memory requirements. Furthermore, analysis substantiates that important error compensation... 

Sometimes the ODEs that arise in studies in nonlinear dynamics can be solved using explicit methods (such as the forward Euler) which require less computations per step and are thus cheaper and ter to implement. The Runge-Kutta femily of algorithms are a popular implementation of the explicit methods. Runge—Kutta methods begin with a Taylor series expansion the order of the particular Runge-Kutta method used is simply the highest order term retained in the Taylor series. 

The only really practical way to solve these equations is by a Monte Carlo simulation which involves the generation of random numbers on a computer. This would come under the definition of level 3 methods and will be discussed further in Section 5.7. The level 2 methods are generally referred to as first order methods because the failure equation for Z is linearly approximated at a point using a Taylor series expansion. Thus... 

Like all other meshless methods, the first step in GFD is to scatter nodal points in the computational domain and along the boimdary. To each node (point), a collection of neighboring nodes are associated which is called star. The number and the position of nodes in each star are decisive factors affecting the finite difference approximation. Particular node patterns can lead to ill-conditioned situations and ultimately degenerated solutions. Using the Taylor s series expansion, the value of any sufficiently differentiable smooth function u at the central node of star, uq, can be expressed in terms of the value of the same function at the rest of nodes, with i = 1,. .N where N is the total number of neighboring nodes in the star and is one less than the total number of nodes in it. In two dimensions, a second-order accurate Taylor series expansion can be written as... 

The use of quantum-chemistry computer codes for the determination of the equilibrium geometries of molecules is now almost routine owing to the availability of analytical gradients at SCF, MC-SCF and CP levels of theory and to the robust methods available from the held of numerical analysis for the unconstrained optimization of multi-variable functions (see, for example. Ref. 21). In general, one assumes a quadratic Taylor series expansion of the energy about the current position... 

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,... 

Compute e. Computers and calculators use series expansions to get quantities such as e, the base of the natural logarithm, and tt, etc. Compute e from a Taylor series expansion. 

If you know the value of fix) at some point x = a, you can use the Taylor series expansion Equation (4.22) to compute fix) near that point ... 

In Chapter 9 w e will use the Euler relationship to establish the Maxwell relations between the thermodynamic quantities. Here we derive the Euler relationship. Figure 5.12 shows four points at the vertices of a rectangle in the xy plane. Using a Taylor series expansion, Equation (4.22), compute the change in a function Af through tw o different routes. First integrate from point A to point B to point C. Then integrate from point A to point D to point C. Compare the results to find the Euler reciprocal relationship. For Af = f(x + Ax,y + Ay) -f(x,y), the hrst terms of the Taylor series are... 

The Poisson-Boltzmann equation (23.5) is a nonlinear second-order differential equation from which you can compute ip if you know the charge density on P and the bulk salt concentration, rioo- This equation can be solved numerically by a computer. However, a linear approximation, which is easy to solve without a computer, applies when the electrostatic potential is small. For small potentials, zeip/kT 1, you can use the approximation sinh(x) s [(I + x) - (I - x)] 12 = X (which is the first term of the Taylor series expansion for the two exponentials in sinh(x) (see Appendix C, Equation (C.l)). Then Equation (23.5) becomes... 

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