Successive substitution convergence

The new guess can be simply the calculated value (this is called successive substitution). Convergence may be very slow because of (1) a very slow rate of approach of to or (2) an oscillation of back and forth around The loop can even diverge. 

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. 

By employing successive substitution of the tear variables and the criterion of Eq. (13-83), convergence was achieved slowly, but without oscillation, in 23 iterations. Computed products are. 

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... 

Equation 12-5 gives an onset temperature Tonset that corresponds to a time-to-maximum rate t (min) using a successive substitution solution procedure. An initial guess of T = 350 K for the right side of Equation 12-5 will give a solution value of Tonset on the left side of Equation 12-5 within 1% or on an absolute basis 3°C. Convergence is reached within several successive substitution iterations. 

This equation can be solved easily by successive substitution. Using a guessed value of the temperature in the first factor on the left of Eq.37a, the exponential term is inverted to obtain an improved value for the guess. When repeated, this procedure converges in just a few cycles for all conditions of practical importance. Eor = 10, the temperature obtained from Eq. 37b is T = 0.04898. The maximum centerline deposition rate can be computed from Eqs. 19 and 22b once the pressure and temperature have been obtained. 

If the rate of chain transfer to DH is significant, then is calculated by the model using successive substitution. Starting with = 1 on the right-hand side, the calculation converges sufficiently with three iterations. If Rm is negligible (for example, when considering chain transfer to monomer only), then <(> can evaluated directly. 

Sometimes successive substitution works very well, converging in a few steps. Three unsatisfactory convergence patterns are also observed from time to time, however. In the first, the successive estimates oscillate about a central value ... 

The second case of slow convergence in successive substitution involves a creeping progression. such as... 

The remedy for this problem is to accelerate the convergence procedure—to jump over many of the intermediate solutions to which continued successive substitution would lead. The... 

The third unsatisfactory convergence pattern is instability. For example, if successive substitution yields a sequence like... 

The preceding five equations may be solved by successive substitution, beginning with the assumption Tq = Tb to initiate a rapidly convergent iteration process. 

For the procedure of successive substitution to be guaranteed to converge, the value of the largest absolute eigenvalue of the Jacobian matrix of F(x) evaluated at each iteration point must be less than (or equal to) one. If more than one solution exists for Eqs. (L.17), the starting vector and the selection of the variable to solve for in an equation controls the solution located. Also, different arrangements of the equations and different selection of the variable to solve for may yield different convergence results. 

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 ... 

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