Successive substitution

In this method, the equation is multiplied by a constant, )3, and an iterative equation is used. 

Under certain conditions, it can be proved that this method converges (Finlayson, 1980). The conditions require a bounded derivative d//dv and also a particular sign of )3. The smaller the value of )3 the slower the convergence to the solution is, but also then it is more likely that a solution can be found. It is sometimes hard to predict a good value of )3. 

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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 ... 

Method of Successive Substitutions Write a system of equations as... 

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. 

Mathematical modehng is essentially the same process, but the description of the component performance is generally much more complex and detailed. This approach enables a user to vary more parameters easier, look into various possibilities for intervention, and predict the response of the system on different influences. Equationsolving does not necessarily have to be done by successive substitution or iteration as this procedure could suggest. 

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. 

HS(G)II0 Seven steps to successful substitution of hazardous substances... 

The bottom-up method uses the same substitution and expansion techniques, except that now, the operation begins at the bottom of the tree and proceeds up. Equations containing only basic failures are successively substituted for higher faults. The bottom-up approach can be more laborious and time-consuming however, the minimal cutsets are now, 1 obtained for every intermediate fault as well as the top event. 

Tliis methodology has also been extended to the use of liquid methylamine/ potassium permanganate (LMA/PP system). When this system is applied to a number of 3-nitro-l,8-naphthyridines (92a-92g), the C-4 position could be successfully substituted by methylamino group yielding 93a-93f. Tire intermediary 4-methylamino-[Pg.305]

The successive substitution of methane hydrogens with chlorine produces a mixture of four chloromethanes ... 

Eq. (18) is now a function of temperature only. It can be solved by any trial and error procedure such as successive substitution or regula falsi (Gerald (1978)). Having obtained the optimum temperature from Eq. (18), the corresponding minimum time can be calculated from Eq. (17). 

However, it may seem that we have ignored the m, n, and p odd combinations that are also possible. For example, in jamesonite, Fe"Pb4Sb5Si4 [80], the mnp values are 034 (or 068), respectively (assuming one Fe(II) = 2 Cu(I)). We have successfully substituted europium into the structure to create a congru-ently melting, semiconducting FePb2Fu2Sb6Si4 [79]. This has a value of A B =... 

At least 90 percent of free-radical-polymerized 2,3-dimethylbutadiene consists of 1,4 units according to ozone degradation experiments. Successive substitution of the methyl groups on carbons 2 and 3 of butadiene is seen to increase the proportion of 1,4 units formed. In polychloroprene no less than 97 percent of the structure consists of 1,4 Cl... 

We found that the negatively charged surfactant, sodium laurel sulfate, can be successfully substituted for the serum proteins used previously. In low ionic strength solutions, the cmc of the surfactant is 8.1 mM [577]. We explored the use of both sub-CMC (data not shown) and micelle-level concentrations. Saturated micelle solutions are most often used at pION. 

Thus, two values can be evaluated, W+ and W, according to the signs in Eq. (137). If the appropriate integrals are known, these quantities can be calculated for a given interatomic distance in the diatomic molecule. As indicated above the successive substitution of the two values W+ and W- yields... 

Successive Substitutions Let fix) = 0 be the nonlinear equation to be solved. If this is rewritten as x = Fix), then an iterative scheme can be set up in the form x,. Fixf). To start the iteration an initial... 

Behavior of successive substitution process Range of optimum q... 

The cyclodimers are liberated from the respective elimination products 8a and 10a via successive substitution processes with incoming butadiene, that regenerates the active catalyst la in an overall exergonic process. For the rate determining reductive elimination step of the C8-channel free-energy activation barriers of 20.1-24.1 kcalmol-1 are predicted for catalysts TTV, that are in excellent agreement with experimental estimates.43 Thus, moderate reaction conditions are required for the catalytic cyclodimerization of 1,3-butadiene.6... 

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