Wegstein convergence method

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 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. 

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 ... 

Thus, Equation 13.36 can be used to accelerate the convergence and is known as the Wegstein method11. If q = 0 in Equation 13.36, the method becomes direct substitution. If q < 0, acceleration of the solution occurs. Bounds are normally set for the value of q to prevent unstable behavior. 

Wegstein method. This does not require a derivative and usually converges more rapidly than the Newton-Raphson. The equation is put in the form... 

Wegstein) was used. Switching to the Broyden method successfully converged to flowsheet. Figure 6.97 gives the temperature and composition profiles in the two reactors. 

The Wegstein and Dominant Eigenvalue methods listed in Figure L.3 are useful techniques to speed up convergence (or avoid non-convergence) of the method of... 

Q.5 < q < 1, direct method diverges in oscillatory manner, Wegstein method converges... 

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 parameter q may be bounded or modified for better convergence control. Wegstein is the default method in many simulators. 

However, sometimes Wegstein or Newton-Raphson methods are employed as convergence acceleration promoters. 

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. 

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