Successive approximations, method

Successive approximation methods, 59 Successive displacements, method of, 61... 

Most of the well-developed methods available for solving such a system falls within the categories of direct or exact-fitted methods and iterative or successive-approximate methods which are gaining increasing popularity. 

A successive approximation method that does not rely on the y-dcpcndcnt part of F being small is the Newton-Raphson method. As seen in Fig. 1, if a value of y, y, gives F x,yx), a closer solution to Eq. (2) can usually be obtained by setting... 

The existing theories adopt, more or less, a successive approximate method in which one first considers segments to be independent of each other and then introduces their connectivity in some way or the other. The first step has been taken not only by the lattice theories but also by the free volume theories (43). However, it is the second step which is not simple and important. While this second step was taken in our theory discussed in the previous section, one must still try to develop an improved treatment of the chain connectivity, avoiding, for example, the product assumption such as Eq. (6.7). 

We can also simplify the solution given in (15.93) using the Picard successive approximation method when we know the initial conditions. If t0 and z0(t0) are the initial conditions and the differential equation is written as z (t) =fit, z), the equation can be written as... 

The condition Vg(ab,0) v enables the application of a successive approximation method to calculate the velocity, concentration and adsorption fields (Dukhin Derjaguin, 1961). Since... 

The use of a successive approximation method, i.e. as a first approximation, the identification of the particle velocity Vp(z,0) with the local liquid velocity v(z,0) seems to be not justified... 

If, as a first approximation, the particle velocity is assumed to be constant when the energy losses are small, the integral in Eq. (11.24) my be evaluated for the case of potential flow. The result can be refined by successive approximation methods and finally takes the form... 

Relativistic effective core potential (RECP) methods, also called relativistic pseudopotential (PP) methods, are probably the most successful approximate methods for the various properties of molecules containing heavy atoms. 

A successive-approximation method was recently suggested by Oblow 47) for the solution of Eq. (2.52) ... 

The Neumann series or the successive-approximation methods provide formulations that, theoretically, should yield the particular solution of Eq. (252). Practical implementation of those methods calls for special care in order to avoid a contribution from the fundamental mode [the term of Eq. (254)]. A fundamental-mode contamination can result, for example, if the source condition of Eq. (253) is not satisfied exactly. Numerical round-... 

In the successive-approximation method [Eqs. (265) and (266)], the external source term appears for every iteration. If this source is localized, a high-order transport method must be applied to all iterations for which the source amplitude is large relative to the term, so as to insure... 

It is also worth considering if the program uses methods that directly solve cubic equations such as (47), which are essential to fit 1 2 and other complex equilibria or if the method of successive approximation is used. Again, neither should be ruled out as the successive approximation method is simpler and faster to implement—however, it may not always converge to solve the equation(s) of interest. 

The value of a is obtained by a similar successive approximation method. 

In the equation (2) it is necessary to determine a value R, for which purpose successive approximation method is appUed. 

Since was calculated according to Equations 4.13 or 4.14, the value of for a non-stretched network was used, therefore one should recalculate this parameter using Such a recalculation results in a somewhat larger value of = 1.48 and = 85 according to Equation 4.15. Then = 0.96 nm and = 6.51 according to Equation 1.8. The precise calculation can be carried out by a successive approximations method, but for our purposes the average value of Z), which is equal to 1.43, is sufficient. The parameters and obtained in... 

The index s represents the initial stationary state. A successive approximation method can be used to solve eq. (15.23) subject to these initial conditions. The initial values of the coefficients c, are introduced in the right-hand side of eq. (15.23). For fc A s, the approximate equations... 

This integral equation can be solved by the successive approximation method (see Sec. 2.8). Let us choose... 

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