Wegstein method

we apply the Wegstein formula to the unstable form of the example  

Find a zero of a nonlinear equation of one variable using the wegstein algorithm. Func is the name of an M-file that contains a real-valued fmction of a single real vaoriable, and x is the initial guess. 

Note that a bad formulation of the problem may cause divergence. 

An optional third argument sets the tolerance for the convergence. The presence of an optional fourth eirgument specifies the maucimum iteration number. 

pre-processing, set default tolerance and maximum iteration /, number. 

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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. 

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... 

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... 

The bounded Wegstein method is the default method in most of the simulation programs. It is a linear extrapolation of successive substitution. 

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

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 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. 

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. 

This defines the nonlinear relationship between heat capacity % and teiq>erature (Eq. 2.3.5). The format is suitable for the Y, Wegstein method and is a divergent expression using direct 7, substution... 

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