The Wegstein Method

A sufficient condition for convergence of Eq. 1.29) to the root x is dial g fjc) I for all X in the search interval. Fig. 1,2b shows the case when this condition is not valid and the method diverges. This analytical test is often difficult in practice. In a computer program it is easier to determine whether x - jTjI jjCj - jc, and, therefore, the. successive x values converge. The advantage of this method is that it can be started with only a single point, without the need for calculating the derivative of the function. 

The Wegstein method may also be used for the solution of the equations of the form 

Starting with an initial value of x, we first obtain another estimation of the root from 

As shown in Fig. 1.3, jCj does not have to be closer to the root than jc,. At this stage, we estimate the function g(x) with a line passing from the points (J i, g(JC )) and (x. g x )) 

It can be seen from Fig. 1.3a that is closer to the root than either Xj and Xj- In iteration we pass the line from the points (xj, (jTj)) and (jc g (jjj)) and again evaluate the next estimation of the root from the intersection of this line with y = x. Therefore, the general iterative formula for the Wegstein method is 

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The Wegstein method is a secant method applied to g x) — x — F x). Numerical Solution of Simultaneous Nonlinear Equations... 

Returning to the example from Figure 13.13, the solution by direct substitution is followed in Table 13.9. If the Wegstein Method is applied after the first two iterations ... 

The Wegstein method is a secant method applied to g(x) = x - Fix). In Microsoft Excel, roots are found by using Goal Seek or Solver. Assign one cell to be x, put the equation for/(x) in another cell, and let Goal Seek or Solver find the value of x that makes the equation cell zero. In MATLAB, the process is similar except that a function (m-file) is defined and the command fzeroCf .xO) provides the solution x, starting from the initial guess xO. 

Programs are written in QUICKBASIC for the Newton-Raphson method with analytical or numerical derivatives and the Wegstein method. The particular equation is entered following line 30. The one used here is from problem PI.02.01. 

Therefore the above expression provides an automatic selection of the damping or accelerating factor c in each iteration. The idea is easy to extend to a system of equations, if each element x of the vector x is regarded to be independent of the others when using the expressions (2.23) and (2.24). Thus the Wegstein method uses separate factors c for each variable ... 

The Wegstein method initially starts out with a direct substitution step ... 

This chapter listed many of the possible units in the Model Library of Aspen Plus. The ammonia process illustrated the procedures (and computer windows) you used to set the process conditions and examine the results. The thermodynamics choices can be verified by comparison with data reported in the literature. Sometimes the calculations do not converge, and then the Wegstein method, or Broyden s method, are useful for accelerating convergence. 

The value of the functions is kept after each iteration in order to make this calculation. The Wegstein method is essentially a secant method, with some constraints on the parameters as described in Chapter 7. It is also possible to use a numerical derivative. 

This defines the nonlinear relationship between heat capacity /, and temperature (Eq. 2.3.11). The format is suitable for the /( Wegstein method and is a convergent expression using direct y, substitution /. 

The Wegstein method converges, even under conditions in which the method of a = (-c) does not. Moreover, it accelerates the convergence when the successive substitution method is... 

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