Taylor series method

The CLQA method for determining the MEP has been compared to the LQA method, the Euler method, and the quadratic and cubic Taylor series methods on an ab initio MCSCF potential energy surface for the reaction. 

All of the methods discussed above rely on computing values of the vector field (i.e. momenta and forces) only. However, as we saw earlier in this chapter, in the setting of a Taylor series method, we may approximate a single step by... 

Similarly, if we expand the function y(A - Ax) around the point x using the Taylor series method, we would obtain the formula... 

Here is another application of the Taylor series method. Suppose that you want to find the values x = x that cause a function to equal zero, fix) = 0. Such values of x are called the roots of the equation. The Newton method is an iteratixe scheme that works best when you can make a reasonable first guess X, and when the function does not have an extremum between the guess and... 

The following section illustrates the Taylor series method, and also introduces an important model in statistical thermodynamics the random walk (in two dimensions) or random flight (in three dimensions). In this example, we find that the Gaussian distribution function is a good approximation to the binomial distribution function (see page 15) when the number of events is large. 

Another important search direction method is the Newton direction. This direction is derived from the second-order Taylor series. Methods that use the Newton direction have a fast rate of local convergence. Nevertheless, the main drawback is that it requires the explicit computation of the Hessian matrix (V /(ar)). 

When discussing derivative methods it is useful to write the function as a Taylor series expansion about the point jc. ... 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

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... 

Interestingly, if one Taylor series expands Eq. (36) and equates the terms of the same order in kj with Eq. (37) one can derive the standard Lagrangian FD approximations (i.e., require the coefficient of kj to be —1, and require the coefficient of all other orders in kj up to the desired order of approximation to be 0.) A more global approach is to attempt to fit Eq. (36) to Eq. (37) over some range of Kj = kjA values that leads to a maximum absolute error between Eq. (36) and Eq. (37) less than or equal to some prespecrfied value, E. This is the essential idea of the dispersion-fitted finite difference method [25]. 

With a few minor modifications, the Gauss-Newton method presented in Chapter 4 can be used to obtain the unknown parameters. If we consider Taylor series expansion of the penalty function around the current estimate of the parameter we have,... 

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... 

In Newton s method for a set of nonlinear equations, each equation is expanded in a truncated Taylor series. The result is a set of linear equations in corrections to previous estimates. Repetition of the process ultimately may converge to correct roots provided initial estimates are sufficiently close. 

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... 

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 ... 

A set of nonlinear equations can be solved by combining a Taylor series linearization with the linear equation-solving approach discussed above. For solving a single nonlinear equation, h(x) = 0, Newton s method applied to a function of a single variable is the well-known iterative procedure... 

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