Approximations successive

A time-honored method of solving complex equations is to select an approximate answer (more or less arbitrarily) to the equation and use the resulting calculation to obtain a better approximation, and repeating this until further change is within the desired limit. The tediousness and time required is reduced almost to zero with the spreadsheet, as will now be illustrated. 

Consider, for example, an equation that describes the pH of a solution of 0.01 M Na2C03 derived by methods outlined in Chapter 4 and here configured specifically for this example  

In this form, we can see that the solution to [H ] is itself a function of F([H ]. The method of successive approximations involves using two columns in the spreadsheet, A for x values, B for values of F([H ]. Place any value of [H ] in A2, then use the formula in the equation above in B2. In A3, use the contents of cell B2, i.e., +B2. Using B2 as source, copy into B3 now, using A3..B3 as source, copy to A4..B10, This table demonstrates how few the number of successive approximations are needed to solve this equation. Notice that when the initial guess — which need not be close — is say 0.001 or 10 , convergence is very rapid you need not spend much time deciding about the first approximation  

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Equation XVII-127 connects the functions 0(F, T), d(Q,P, T) and f Q) and, in principle, if any two are known or can be assumed, the remaining one can be calculated. As may be imagined, many choices of such pairs of functions have been examined, often designed so that Eq. XVII-127 can be handled analytically alternatively, various schemes of successive approximations may be used. The field has become somewhat of a happy hunting ground for physical chemists and there are numerous reviews of the now-extensive literature (see Refs. 144-147 the last is a personalized account). For this reason only some generic approaches will be discussed here. 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

MacDonald J K L 1933 Successive approximations by the Rayleigh-Ritz variation method Phys. Rev4Z 830-3... 

The family of hierarchical elements are specifically designed to minimize the computational cost of repeated computations in the p-version of the finite element method (Zienkiewicz and Taylor, 1994). Successive approximations based on hierarchical elements utilize the derivations of a lower step to generate the solution for a higher-order approximation. This can significantly reduce the... 

Olivieri, A. G. Solution of Acid-Base Equilibria by Successive Approximations, /. Chem. Educ. 1990, 67, 229-231. 

Integration is by successive approximation, using essentially Eulers method. For the first trial. 

The potentiometer principle is used in the successive approximation instrument, where the unknown voltage is compared with a succession of known voltages until balance is obtained. The total time for measurement and display is about 5 ms. 

In practice, an initial guess, po x), is made and successive approximations to p x) are obtained by iterating equation 4.83. 

In most of the problems you will work, the approximation a — x a is valid, and you can solve for [H+] quite simply, as in Example 13.7, where x = 0.012a. Sometimes, though, you will find that the calculated [H+] is greater than 5% of the original concentration of weak acid. In that case, you can solve for x by using either the quadratic formula or the method of successive approximations. 

The method of successive approximations. In the first approximation, you ignored the x in the denominator of the equation. This time, you use the approximate... 

The second answer is physically ridiculous the concentration of H+ cannot be a negative quantity. The first answer is the same one obtained by the method of successive approximations. 

Substances, 255q density ofj 14 15 properties of, 13-19 solubility of, 15-17 Substitution reaction, 603 Successive approximations A technique... 

To make matters worse, the use of a uniform gas model for electron density does not enable one to carry out good calculations. Instead a density gradient must be introduced into the uniform electron gas distribution. The way in which this has been implemented has typically been in a semi-empirical manner by working backwards from the known results on a particular atom, usually the helium atom (Gill, 1998). It has thus been possible to obtain an approximate set of functions which often serve to give successful approximations in other atoms and molecules. As far as I know, there is no known way of yet calculating, in an ab initio manner, the required density gradient which must be introduced into the calculations. 

A similar approximation scheme, which gives successive approximations, in first order, to afl can be developed the thermal conductivity for the central force law is ... 

A method of successive approximation (s.a. method) is one that would provide the solution as the limit of an infinite sequence of steps if these steps were carried out exactly. These steps are usually quite simple, and nearly identical, each to the next, so that programming is a relatively easy task. Most methods operate upon a given approximation to obtain a better one, hence they are self-correcting. Because of rounding, at some stage the computed correction will no longer be... 

Models are usually formed through a process of successive approximations, i.e., refinements are introduced successively, based upon the worth and reliability of additional information and upon the suitability of the model for prediction. This refinement process may be described as follows 3... 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

Successive approximation methods, 59 Successive displacements, method of, 61... 

FIGURE 4 Successive approximations to the true tangent are obtained as the two points defining the straight line come closer together and finally coincide. 

By repeating the calculation for various values of x, one can obtain y and Sx as functions of x and find the minimum of the latter by successive approximations. The value of x at this minimum (xo) gives the estimate of the isokinetic temperature Xo The corresponding values yo and So are obtained from eqs. (52) and (53) So has... 

Because of the convenient mathematical characteristics of the x -value (it is additive), it is also used to monitor the fit of a model to experimental data in this application the fitted model Y - ABS(/(x,. ..)) replaces the expected probability increment ACP (see Eq. 1.7) and the measured value y, replaces the observed frequency. Comparisons are only carried out between successive iterations of the optimization routine (e.g. a simplex-program), so that critical X -values need not be used. For example, a mixed logarithmic/exponential function Y=Al LOG(A2 + EXP(X - A3)) is to be fitted to the data tabulated below do the proposed sets of coefficients improve the fit The conclusion is that the new coefficients are indeed better. The y-column shows the values actually measured, while the T-columns give the model estimates for the coefficients A1,A2, and A3. The x -columns are calculated as (y- Y) h- Y. The fact that the sums over these terms, 4.783,2.616, and 0.307 decrease for successive approximations means that the coefficient set 6.499... yields a better approximation than either the initial or the first proposed set. If the x sum, e.g., 0.307,... 

In Fig. 4.4, the starting points (initial estimates) are denoted by circles in the A, B plane the evolution of the successive approximations is sketched by arrows that converge on the optimum, albeit in a roundabout way. The procedure is robust no local minima are apparent the minimum, marked by the square, is rapidly approached from many different starting points within the given A, B plane (black symbols) note that a B-value > 0, an A-value < 0, and combinations in the lower left comer (gray symbols) pose problems... 

Most of the well-developed methods available for solving such a system falls within the categories of direct or exact-fitted methods and iterative or successive-approximate methods which are gaining increasing popularity. 

Iterative methods of successive approximation are in common usage for rather complicated cases of arbitrary domains, variable coefficients, etc. Throughout the entire section, the Dirichlet problem for Poisson s equation is adopted as a model one in the rectangle G = 0 < x < l, a = 1,2 with the boundary P ... 

As the concentrations of dissolved water and EDC are small, the best approach to this problem is by successive approximation rather than by setting up and solving equations for the unknown concentrations. 

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