Numerical methods iterative method

I will return to this diagram near the end of the chapter, particularly to amplify the meaning of error removal, which is indicated by dashed horizontal lines in Fig. 7.1. For now, I will illustrate the bootstrapping technique for improving phases, map, and model with an analogy the method of successive approximations for solving a complicated algebraic equation. Most mathematics education emphasizes equations that can be solved analytically for specific variables. Many realistic problems defy such analytic solutions but are amenable to numerical methods. The method of successive approximations has much in common with the iterative process that extracts a protein model from diffraction data. 

Once some idea has been obtained of the type of expression that is required, it is then possible to integrate the rate equations numerically. An iterative method is often convenient the iteration would normally take the exponents in the various concentration terms as fixed and would proceed by substituting trial values of the rate coefficients (estimates of these having been obtained from prior experiments) in the differential equations and integrating in this way, it is possible to find values which give a satisfactory match of calculated and observed data. The question of the criterion of fit is outside the scope of this chapter. 

Unlike (2.166), the constructed iterative equation (2.167) is linear, which allows us to apply the standard numerical methods to solve it. 

I hese equations cannot be used directly, and numerical methods are needed to compute the velocity components. The velocity components can be found by implicit differentiation and using an iterative technique.-" ... 

The set of equations in the previous section 4 is solved numerically using the NIM (natural iteration method). When we use the constraint Lagrange multipliers, the NIM consists of the "Major Iteration" and the "Minor Iteration". The major iteration solves the set of basic of equations in (11), the reduction relations in (2) and the normalization relation. It was proved that this iteration always converges from whatever initial guess values it starts with. 

From Pm still further information about the parameters of the desorption process can be obtained. To this end, Eq. (8) must be solved. The solution, however, is accessible only in the case of desorption alone. If the contribution of the second term in Eq. (8) is appreciable, it is necessary to insert for P from Eq. (13). Thus, nonlinear differential equations result even for the most simple cases (x = 1, or the equilibrium desorption), which can be solved by numerical methods only or by iterative methods provided the second term in Eq. (8) is small. 

Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992, Numerical Recipes in C, Cambridge University Press, 2nd edition Richardson, W.H., 1972, Bayesian-based iterative method of image restoration, JOSA 62, 55... 

In this chapter economical direct and iterative methods are designed for numerical solution of difference elliptic equations. 

On computational stability of iterative methods. Until recent years the iterative method with optimal set of Chebyshev s parameters was of little use in numerical solution of grid equations. This can be explained by real facts that various sequences turn out to be nonequivalent in computational procedures. 

Since the modified iterative method is completely numerical, data can be used directly from the monodisperse chromatograms to characterize the axial dispersion, eliminating the need for a specific axial dispersion function. The monodisperse standards were used to represent the spreading behavior for particle ranges as given in reference (27). 

To determine cos one should solve the set of f integral equations for probabilities of degeneration u 0(r),...,u f 1 (r) and substitute these functions into functional 0) [u] ( q. 62). Numerical solution of these equations by means of the iteration method presents no difficulties since the integral operator is a contrac-... 

In this section we shall review the numerical methods used for solving Eq. (53). These are chiefly iterative methods. They differ in the procedures according to which the iterates are generated. 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

Prior to the advent of high-speed computers, methods of optimization were limited primarily to analytical methods, that is, methods of calculating a potential extremum were based on using the necessary conditions and analytical derivatives as well as values of the objective function. Modem computers have made possible iterative, or numerical, methods that search for an extremum by using function and sometimes derivative values of fix) at a sequence of trial points x1, x2,. 

To carry out an iterative method of numerical minimization, start with some initial value of x, say jc° = 0, and calculate successive values off(x) = x2 — 2x + 1 and possibly df/dx for other values of x, values selected according to whatever strategy is to be employed. A number of different strategies are discussed in subsequent sections of this chapter. Stop when/(jt +1) — /( ) < ex or when... 

Because solving the transcendental equation (1.5.23) requires iterative methods of root finding, the numerical solution to this equation is postponed to Section 3.1. o... 

Digital simulation is a powerful tool for solving the equations describing chemical engineering systems. The principal difficulties are two (1) solution of simultaneous nonlinear algebraic equations (usually done by some iterative method), and (2) numerical integration of ordinary differential equations (using discrete finite-difference equations to approximate continuous differential equations). 

The simultaneous solution of eqns. (72) and (79) when h is not zero is generally achieved by a numerical method which considers small increments in reactor volume and then iterates the calculation of the resulting temperature and fractional conversion in a manner similar to that described for Sect. 2.5.3 for a batch reactor. Cooper and Jeffreys [3] give an illustrative example, together with a computer flow diagram, for calculating the reactor volume. 

To solve the age from the slope requires numerical method (such as trial and error, iteration, etc.). 

Section III discusses briefly (1) the relation between our logical equations and the differential equations as used in chemical kinetics (2) some aspects of logical versus numerical iteration methods (3) the possible application of our method (initially developed for genetic purposes) to other fields, and more particularly chemical kinetics and (4) the possibility of using this method al rovescio, that is, in a synthetic (inductive) way. In this perspective we assume that the essential elements of a system have been correctly identified and we ask to what extent one can proceed rationally from the observed behavior toward sets of interactions which account for this behavior. 

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