Wegstein algorithm

Most simulation programs have a block, called a convergence block, that performs such calculations using the Wegstein algorithm. The output stream from this block contains the assumed set of tear stream variables, and the input stream contains the values calculated by working around the cycle. The block diagram for the simulated process would appear as follows ... 

You are to write the code for a convergence module that can deal with one to three tear stream variables using the Wegstein algorithm, as outlined in Appendix A.2. The object is to determine the values of one, two, or three of the variables x, xi, and X2 that satisfy the relations... 

Assumed values of x, (initially guessed, subsequently calculated by the Wegstein algorithm). 

Calculate the next value of x using the Wegstein algorithm. 

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. 

Wegstein Algorithm. If the equations to be solved have the form of Equation A.2-10 [i.e., X, = fi(x, X2,. ., x )], guess values for all n variables and apply the procedure of Section A.2g separately to each variable. Terminate when the convergence criteria are satisfied for all variables. 

Wegstein algorithm. The same functionsare used togeneratecalculatedvaluesol.v andv from ... 

For the solution of several equations simultaneously, each x is treated independently, a procedure that may possibly cause some instability if the x s interact. In such cases, an upper limit should be placed on (, say 0 t 1. The Wegstein algorithm is for stage k. 

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. 

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. 

The M-file cp3.m (Figure 2.19) has the convergent path formulation of equation (2.3.11). It can be used with the M-file wegstein.m to obtain solutions to the problem. The algorithm converges rapidly and from all initial guesses for the temperature T. 

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