Successive substitution iterative method

Let X be the guess for the tear stream variables at the A th iteration. Let F(X ) be the calculated result for that tear stream s values. Treat each variable with a secant method. Do two or more successive substitution iterations to generate F(X ) = X k Then accelerate ... 

By applying Aitken s method to a linearly convergent sequence obtained from fixed point (successive substitution) iteration, we can accelerate the convergence to quadratie order. This procedure is known as the Steffenson s method, which leads to Steffenson s algorithm as follows. 

Some formulas, such as equation 98 or the van der Waals equation, are not readily linearized. In these cases a nonlinear regression technique, usually computational in nature, must be appHed. For such nonlinear equations it is necessary to use an iterative or trial-and-error computational procedure to obtain roots to the set of resultant equations (96). Most of these techniques are well developed and include methods such as successive substitution (97,98), variations of Newton s rule (99—101), and continuation methods (96,102). 

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 ... 

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

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 ... 

Once the tear streams are identified and the sequence of calculations specified, everything is in order for the solution of material and energy balances. All that has to be done is to calculate the correct values for the stream flow rates and their properties. To execute the calculations, many computer codes use the method of successive substitution, which is described in Appendix L. The output(s) of each module on interation k is expressed as an explicit function of the input(s) calculated fi om the previous iteration, A - 1. For example, in Fig. 5.16 for module 1,... 

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... 

Iterative methods are used to solve such equations, and the successive substitution and... 

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 ... 

Nonlinear Volterra eqnations are solved by iterative methods. Two iterative methods are presented here. One is Newton s method, and the other is the method of successive substitution. 

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