Wegstein iterative method

Another term used when discussing iteration methods is the tear stream. This is the stream that is guessed (or set to zero) when stepping through the process the first time. In Figure 7.16, if you compute in the order mixer, reactor, separator, then stream 6 is the stream you must assume and check - it is the tear stream. If you compute in the order reactor, separator, mixer, then stream 2 is the tear stream. You apply the Wegstein iteration method to the tear stream. Of course, if the calculations are simple, as would be the case if all physical properties are specified and there are no energy balances, then you can use a spreadsheet. It may be so fast that you do not care whether 10, 100, or 1000 iterations are needed. 

Another combination method shown to be effective is the delayed Wegstein method (59). Here Wegstein s method is applied at fixed intervals between direct iterations. Orbach and Crowe (71 ) used the Dominant... 

In this case, Wegstein s method did not accelerate the convergence. In fact, the large jump in the value of x in iteration 3 (when the Wegstein procedure was first used) could haN e been the first symptom of an instability, but the algorithm recovered well. 

Wegstein s method, which is used in many flowsheeting codes, accelerates the convergence of the method of successive substitutions on each iteration. In the secant method, the approximate slope is... 

Wegstein. This method uses a parabolic extrapolation such that the updated values are computed from two previous iterations, with the formula ... 

Wegstein s method can be employed to accelerate convergence when the method of successive substitutions requires a large number of iterations. As shown in Figure 4.11b, the previous two iterates of/ x and x are extrapolated linearly to obtain the next value of x as... 

Method ofWegstein This is a variant of the method of successive substitutions which forces and/or accelerates convergence. The iterative procedure Xi + = F xC) is revised by setting x + i = F xi) and then taking Xi + = qxi -i- (1 — q)xi + i, where is a suitably chosen number which may be taken as constant throughout or may be adjusted at each step. Wegstein found that suitable q s are ... 

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 ... 

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 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. 

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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