Wegstein

Two values of arc made and xj. The corresponding two values of x are calculated x and x 2, The equation of a straight line joining these two points is 

to calculate a new guess xl, we want to find where this straight line intersects the 45 line (where = x = x ). 

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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 ... 

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

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. 

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 ... 

Wegstein JH (1958) Accelerating Convergence of Iterative Processes, Commun Assoc Comp Mach, 1 9. 

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. 

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

However, to test the Newton-Raphson and Wegstein methods, starting values were assumed -1 and +1. The successive evaluations are tabulated. Wegstein is a little faster for the negative root but a little slower for the positive one. 

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 ... 

Example 2.3.1 Equilibrium of chemical reactions by Wegstein method... 

Derive the formulas (2.23) and (2.24) of the Wegstein iteration from the geometrical idea. 

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. 

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... 

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