Wegstein’s method

This would suggest that initial approximations to J (X) using Wegstein s method should be followed by using Broyden s method. 

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... 

It is not difficult to show that Wegstein s method is tantamount to generating two points on the curve of /(x) versus x and determining as the next estimate the intersection of the line between these two points and the 45° line [at which x = /(x)]. 

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... 

Broyden s method and Wegstein s method are two examples of this approach that use different approximations for In the former, which is based... 

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... 

Figure 4.11 Convergence of a recycle loop (a) successive substi- tution method (b) Wegstein s method, i...

Thus, weights q and 1 — q are applied, respectively, to x and/(jc ). Equation (4.10), withq defined by the slope, is usually employed when the slope is less than 1, such that q < 0. Typically, q is bounded between —20 and 0 to ensure stability and a reasonable rate of convergence. Wegstein s method reduces to the method of successive substitutions, x = / x ), when = 0. 

Wegstein s method obtains the relaxation factor by applying a secant method independently to each component of x. Then from each component, we have... 

In practice, DEM and Wegstein s methods suffer from instability problems, since large acceleration factors are encountered in most problems. To stabilize the problem, it is common to use only an acceleration each 2-8 direct substitution steps and to bound the acceleration factor as follows ... 

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. 

When we use a minimum number of tear streams and the set is redundant, we observe that the redundancy produces some instabilities in the convergence of both Wegstein s and DEM methods (see Figure 8.16b). This instability is not observed in nonredundant sets (Figure 8.16c). Even though the number of tear variables (and total variables) is large in scenario 2, the convergence is considerably faster when the redundancies are minimized. 

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 ... 

Three methods that can be used to find the values of xj,..., that satisfy n simultaneous equations are extensions of methods given previously for single-variable problems. They are (a) successive substitution, (b) the Wegstein algorithm, and (c) the Newton-Raphson method (a multivariable extension of Newton s rule). The example that concludes this section illustrates all three algorithms. 

In the previous subsection, the successive substitution and Wegstein methods were introduced as the two methods most commonly implemented in recycle convergence units. Other methods, such as the Newton-Raphson method, Broyden s quasi-Newton method, and the dominant-eigenvalue method, are candidates as well, especially when the equations being solved are highly nonlinear and interdependent. In this subsection, the principal features of all five methods are compared. 

---