Repeated substitution

A very large number of methods of solving systems of nonlinear algebraic equations has been devised (Ortega and Rheinbolt, 1970). However, just two methods are employed in the algorithms presented in this book repeated substitution and Newton s method we review these methods below. 

To use repeated substitution, the equations to be solved must be expressed in the form 

Here the only unknown is the value of the vector (x). Thus, by solving the linear system of Eqs. C.2.5 it is possible to obtain a value for this vector. If the new vector (x) obtained in this way does not actually satisfy the set of equations (F) then the procedure can be repeated using the calculated value as a new (Xq). The entire procedure is summarized in Algorithm C.2. 

Check for convergence. If (x + j - (x is less than some prescribed small number, stop. Otherwise, increment k and return to Step 2. 

Check for convergence if not obtained, increment k and return to Step 2. 

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The entire theory of Section 5.2 may now be repeated, substituting S, E, and K for s, e, and k, respectively. Obviously, the results will parallel those of Section 5.2, with referential variables in place of the small deformation variables. Rather than repeat the development in Section 5.2, the results may be obtained by substituting majuscules for minuscules in the salient equations. The stress relation (5.3) becomes... 

Table 22.1), which has been modified by appropriate substitutions to yield the desired molecule. Thus, aliphatic hydrocarbons can be built up from methane by repeated substitutions of methyl groups for hydrogen atoms. Other compounds are formed by substitution of functional groups for CHn groups. The heat capacity constants are those for a cubic polynomial in the temperature, which are similar to those discussed in Chapter 4. 

In hot-water-soluble, carob galactomannan, the relatively high level of couplets of neighboring D-galactosyl groups would reflect a capacity for a repeat substitution on the opposite side of the mannan chain, once the steric hindrance involved in formation of the enzyme-substrate complex has been... 

Fig. 4.6. Coaxial NMR tubes for the measurement of magnetic susceptibility of a paramagnetic solute. Solution A contains the paramagnetic solute and an inert probe substance. Solution B contains the probe substance in the same solvent. The measurement is then repeated substituting the paramagnetic solute in A with a diamagnetic analog of the same concentration.

Repeat, substituting iodine for bromine, and note that the deep violet color, which the free iodine imparts to the globule of carbon disulphide, is likewise bleached on shaking the globule with chlorine water. 

To obtain the steady state(s) of Equation (5.43), we have by repeated substitution ... 

Error = 100 x (experimental value - trae value) / (true value) Average % Error = (Z1% Error for 16 amino acidsi) /16 These calculations were repeated substituting the true value with analyzed value . Analyzed values are the overall average determined by all the participants and may represent a more realistic figure than the theoretical values, which were determined by protein or nucleic acid sequencing. These values were used to compare the errors of the pre-hydrolyzed and protein samples. In analyzing data, we excluded sites with errors >3 standard deviations from the mean. 

In so far as [B] (or, more generally, [D] ) depends on the composition of the mixture and as the composition is, in turn, a function of position 17, we may regard [T ] as a function of 77. Thus, Eq. 8.3.50 is a first-order matrix differential equation of order n — 1 with a variable coefficient matrix [ (17)]. This equation may be solved by the method of repeated substitution as shown in Appendix B.2. The solution is... 

Algorithm 8.4 Algorithm Based on Repeated Substitution for Calculation of Mass Transfer Rates from Solutions of the Linearized Equations... 

If the total flux Af is not specified, an iterative approach is required for evaluation of the molar fluxes A -. A procedure based on repeated substitution is provided in Algorithm 8.7. A still more efficient procedure can be devised using Newton s method (cf. Algorithm 8.5) (Krishna and Taylor, 1986). 

It is worth noting that simple repeated substitution of the fluxes is not effective for solving this particular problem if the calculations are started with 3/eff = 1 (corresponding to a null estimate of the fluxes). The oscillations in the fluxes that result with simple repeated substitution can be avoided by using an average of the last two computed estimates of the fluxes in the evaluation of the mass transfer rate factors. In this case, however, we used Newton s method to solve a single function of the total flux (cf. Algorithm 8.5). 

The computational methods discussed above for use with the ideal solution simplifications of the general relations presented could also be employed here. We recommend the use of repeated substitution to calculate the from Eqs. 8.7.4 and Newton s method for... 

It is necessary to use an iterative method to compute the flux from Eq. 9.2.15. Repeated substitution of the fluxes, starting from an initial guess calculated with 3 = 1, will usually converge in only a few iterations. 

This is the implicit vapor pressure equation. It is solved by repeated substitution. Beginning with an estimated p, Vy and are calculated at a given T until they converge adequately. 

The saturated volumes Vy and in Equation (4.468) are solved in the RK eos. Equation (4.467), at the vapor pressure, which makes the equation implicit. The equation can be readily solved for the saturated pressure by repeated substitution. Agreement with vapor pressure is quahtative at best. 

This equation is implicit in fly, inasmuch as the saturated volumes are dependent on fly, but the equation can be readily solved, for instance, by the method of repeated substitution. By using the solved value of fly in the eos. Equation (4.162), the vapor pressure calculated by the eos simply reproduces the experimental vapor pressure data. Wilson correlated the fly s that are fitted to vapor pressure data of a number of normal fluids to obtain Equation (4.157), the Wilson eos. Vapor pressure calculated by the Wilson eos is improved over that of the RK eos, but the accuracy still leaves something to be desired. Soave correlated the vapor pressure, fitting fly with Equation (4.164). Even better, the Soave eos is useful for the quantitative calculation of vapor pressure. In addition, the Peng-Robinson and the chain-of-rotators eos s provide quantitative calculations of vapor pressure. 

Equation (15-36) is seen to be a recurrence formula which gives the error vector at the end of the k -f 1st iteration in terms of the error vector for the kth iteration. By repeated substitution of the vector Ek into the expression for Ek+1,... 

Heat in the flame of the burner a small piece of sulphur in a deflagrating spoon, and when it has begun to burn with a small blue flame, lower it into a bottle of oxygen. (7) What effect has oxygen on burning sulphur (8) What is formed Repeat substituting a little red phosphorus for the sulphur. 

A repeated substitution leads to an expression of this free radical concentration as a function of the concentration of radicals being composed of one monomeric unit, [Ri ]. 

Repeated substitution of (29) into itself yield the series expansion... 

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