Successive substitution convergence method

Successive Substitutions Let/(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F x), then an iterative scheme can be set up in the form Xi + = F xi). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writings = F x), of which there will usually be several forms. However, if 7 is a root of/(x) = 0, and if IF ( 7)I < I, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xi + is proportional to the first power of the error in xi for large k. 

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 equation in cell B1 is copied into cells Cl though El. Then turn on the iteration scheme in the spreadsheet and watch the solution converge. Whether or not convergence is achieved can depend on how you write the equations, so some experimentation may be necessary. Theorems for convergence of the successive substitution method are useful in this regard. 

With the SR method, convergence is often rapid even when successive substitution of Z) and Vj is used from one iteration to the next. 

These values of CA and CB are used for the rest of the calculations. It may be noted that the method of successive substitution used here is not the best or the most reliable method of iteration. In this particular example however, the convergence is rapid leading to final values of ... 

Successive substitution is the simplest although generally the least effective method for performing these calculations. Values are assumed for one or more of the unknown variables other variables are then determined from some of the equations new values of the assumed variables are determined from the remaining equations. The process is repeated until convergence is obtained. For flue gas desulfurization examples, many of the variables are highly constrained, and the calculation sequence can easily move into infeasible regions. The solution sequence frequently oscillates. 

Another error function used in converging on a correct solution by a method of successive substitution involves K-factors. The Kj for the mixture are determined from the fugacity coefficients with Equation 15-23. Then... 

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... 

Convergence proofs are available under certain conditions (Finlayson, 1980), and once the iterate value gets close to the solution, the convergence is very rapid. This method is generally better than the successive substitution method, except for special cases, but sometimes a good initial guess is required. 

The convergence of the successive substitution method is slow, i.e., it may require many iterations for the sequence to converge. The Newton-Raphson method has a faster rate of convergence, which is given as follows ... 

Fig. 3.16 Residual history of the successive substitution method. The convergence depends highly on the relaxation factor D. For D=0.1, oscillations occurred.

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