Iterative formula

Regarding x3 = x 0 as a new starting value, repeat to form x3 = x"Q and continue. It can be shown that if the initial iteration is of order k, that of the iteration that produces x0,x 0,Xq, is of the order 2k — 1. Evidently the transformation could be applied again, using the terms x0, x 0, and x2 of the derived sequence to produce the initial term of a new derived sequence. However, new sources of rounding error are introduced in this process, and the finally accepted approximation should be a result < ( ) of substituting a into the basic iteration formula (2-17). 

For any of the three procedures outlined in this section, in minimization you assume the function is unimodal, bracket the minimum, pick a starting point, apply the iteration formula to get xk+l (or jc ) from xk (or xP and xP), and make sure that fixk+l) [Pg.161]

In Equation 44, the intrinsic viscosity is known implicitly and can be approximated by the Newton-Raphson iteration technique (35). The iteration formula is... 

Using least squares statistical techniques, the relationship between the second and third virial coefficients with molecular weight was obtained. This was necessary to make kinetic light-scattering measurements at a constant concentration. An iteration formula was given to calculate Mw from the lima o Kc/Re value at one concentration. 

According to iterative formulae (7.26), (7.27), the fii st iteration of the inverse problem solution is given l)y the crxpression... 

The Zk T) integral can be computed by means of the following iterative formula ... 

Again, luck allowed Robert May to discover a chaotic behavior in the numerical simulation of Verfmlst s logistic equation. As shown in Appendix C, May proposed a numerical equivalent of the logistic equation where the peculation of a given generation (n+1) is related to the populaticm of the previous generation (n) by the iterative formula ... 

The explicit relation i = i(AE) can be determined numerically starting from the implicit function (11), which represents the central problem of the foregoing discussion, and using the Newton method [56], which, as everyone knows, is based on the iterative formula... 

This procedure has been called PCM-CLSn (the acronym CLS reflects the partial closure of the iterative formulas we have exploited). 

Conversely, if both of the elasticity modulus of aggregate and concrete are given, the mortar elasticity modulus can be obtained by the inverse function of Mori-Tanaka formula. Because it is difficult to obtain the explicit expression of inverse function for mortar bulk modulus and shear modulus, iterative computation becomes a good choice. Eqs. 8 and 9, which are the iterative formulas to obtain mortar bulk modulus and shear modulus respectively, are transformed by the simplification and deduction of Eqs. 1-4 ... 

To solve eq. (5.3-3la) for the reduced spreading pressure we need to resort to a numerical procedure. An effective tool to meet this goal is the Newton-Ralphson method, which is an iterative method to obtain z. The iteration formula for the reduced spreading pressure is given below ... 

The iteration formula for the solution of the hypothetical pressure is then ... 

Applying the Newton-Raphson formula to eq. (5.3-57), we obtain the following iteration formula for the reduced spreading pressure... 

The iteration formula for the pure component pressures obtained by the application of the Newton-Raphson method to eq. (5.4-9a) is... 

Having obtained the vector 5 the iteration formula (5.4-14) can be executed... 

Like any iteration formula, the rate of convergence depends on a good choice of the initial guess. In this section, a choice of the initial guess is suggested and this is based on the behaviour of the isotherm at low pressures. We illustrate this with the O Brien and Myers equation. 

The iteration formulas for finding the solutions for Nj and N2 are (by application of the Newton-Raphson method) ... 

The iteration formula for the flux vector N of dimension (n-1) is simply... 

We let the function f to denote eq.(8.7-33), then the iteration formula for the Newton-Raphson method to find y is... 

Fiiiite Differences. Many of the equations used in numerical analysis contain ordinary or partial derivatives. One of the most important techniques used in numerical analysis is to replace the derivatives in an equation, or system of equations, with equivalent finite differences of the same order, and then develop an iterative formula from the equation. For example, in the case of a first-order differential equation with an initial value condition, such as/(x) = F[x,/( )],... 

This technique is known as the Newton-Raphson formula or Newton s iteration formula. 

In [164] the Variational Iterative Method is reconsidered for initial-value problems in ordinary or partial differential equations. A reconsideration of the Lagrange Multiplier is proposed. The above reconsideration is taken place in order the iteration formula and the convergence analysis to be simplified and facilitated. 

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