Method of successive

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... 

Unfortunately, fragmentation by construction does not appear to be independent of the random construction algorithm. An alternative method of successive segmentation (Grady and Kipp, 1985) of the surface, as illustrated in Fig. 8.24 leads to a fragment distribution which agrees well with a linear exponential distribution ((8.59)) and differs significantly from the Mott distribution ((8.58)). 

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. 

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... 

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... 

Banachiewicz method, 67 characteristic roots, 67 characteristic vectors, 67 Cholesky method, 67 Danilevskii method, 74 deflation, 71 derogatory form, 73 "equations of motion, 418 Givens method, 75 Hessenberg form, 73 Hessenberg method, 75 Householder method, 75 Jacobi method, 71 Krylov method, 73 Lanczos form, 78 method of modification, 67 method of relaxation, 62 method of successive displacements,... 

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 ... 

Matheson, I. B. C. The method of successive integration a general technique for recasting kinetic equations in a readily soluble form which is linear in the coefficients and sufficiently rapid for real time instrumental use. Anal Instrum. 1987, 16, 345-373. 

This can be solved with the quadratic equation, and the result is = 0.0118 moles. We can attempt the method of successive approximations. First, assume that 0.0240. We obtain ... 

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 ... 

Words that can be used as topics in essays 5% rale buffer common ion effect equilibrium expression equivalence point Henderson-Hasselbalch equation heterogeneous equilibria homogeneous equilibria indicator ion product, P Ka Kb Kc Keq KP Ksp Kw law of mass action Le Chatelier s principle limiting reactant method of successive approximation net ionic equation percent dissociation pH P Ka P Kb pOH reaction quotient, Q reciprocal rule rule of multiple equilibria solubility spectator ions strong acid strong base van t Hoff equation weak acid weak base... 

In future, Mulliken and others would attempt to substitute ab initio methods for the earlier methods of successive approximations. With this, quantum chemistry was to become more abstract, more theoretical, than in its initial development. The new approaches seemed to express increasing confidence in a deep level of understanding, the hubris so characteristic of the quest for mathematical certainty. There were dangers in this which Coulson warned might deceive the theoretician. "One is almost tempted to say. .. at last I can almost see a bond. But that will never be, for a bond does not really exist at all it is a most convenient fiction which, as we have seen, is convenient both to experimental and theoretical chemists." 151... 

To evaluate log K", it is necessary to know Ae and, therefore, C,. Yet to know C, we must have a value for a, which depends on a knowledge of Ag. In practice, this impasse is overcome by a method of successive approximations. To begin, we take Ae = Ao and make a first approximation for a" from Equation (20.26). With this value of a", we can calculate a tentative C which can be inserted into Equation... 

For solving equation (1) we use the standard method of successive approximations, putting... 

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