Numerical techniques convergence

Numerical techniques are sometimes required in the solution of thermodynamics problems. Particularly useful is an iteration procedure that generates a sequence of approximations which rapidly converges on the exact solution of an equation. One such procedure is Newton s method, a technique for finding a root X = Xr of the equation... 

The second part of training radial basis function networks assumes that the number of basis functions, i.e., the number of hidden units, and their center and variability parameters have been determined. Then all that remains is to find the linear combination of weights that produce the desired output (target) values for each input vector. Since this is a linear problem, convergence is guaranteed and computation proceeds rapidly. This task can be accomplished with an iterative technique based on the perception training rule, or with various other numerical techniques. Technically, the problem is a matrix inversion problem ... 

This point is very important in the case of computerized systems for both the performing of polarization curves and the processing of experimental data without an operator. The success of the application of some numerical analysis techniques [34, 37] depends on the absence of problems concerning the convergence of numerical sequences used by the method adopted. Such problems may arise when the interval width of the potential difference AE is so small that the available experimental data do not contain the information required for a correct use of the numerical technique. In this case, the evaluation of the electrochemical parameters /<, a and by other methods not subject to convergence criteria is, in principle, physically unacceptable because in the region examined the law (2) cannot be deemed valid. This particular problem has been dealt with by the... 

Still, the linear and nonlinear CG methods play important theoretical roles in the numerical analysis literature as well as practical roles in many numerical techniques see the recent research monograph of Adams and Nazareth for a modem perspective. The linear CG method, in particular, proves ideal for solving the linear subproblem in the truncated Newton method for minimization (discussed next), especially with convergence-accelerating techniques known as preconditioning. 

This transformation usually improves the convergence and stability of the numerical techniques that are used in nonlinear regression programs. Let s choose To as the midpoint of the range of temperatures in Table 6-5, i.e., 7b = 90°C = 363 K. We wiU use nonlinear regression to find the values of 363) and E. 

From this discussion it can be concluded that the route to obtaining converged solutions to the set of semiconductor device equations in this BV problem is not to be found in searching for some improved numerical techniques that can handle extremely large solution derivatives at the boundaries, but in modifying the boundary conditions so that known semiconductor physical limits are not exceeded. A more realistic set of boundary conditions on the minority carriers at the boundaries of this problem is needed. This means the boundary condition for on the left boundary and on the right boundary. Without much discussion an acceptable set of boundary conditions can be formulated as ... 

Most flow sheets have one or mote recycles, and trial-and-ettot becomes necessary for the calculation of material and energy balances. The calculations in a block sequential simulator ate repeated in this trial-and-ettot process. In the language of numerical analysis, this is known as convergence of the calculations. There ate mathematical techniques for speeding up this trial-and-ettot process, and special hypothetical calculation units called convergence, or recycle, units ate used in calculation flow diagrams that invoke special calculation routines. 

There are special numerical analysis techniques for solving such differential equations. New issues related to the stabiUty and convergence of a set of differential equations must be addressed. The differential equation models of unsteady-state process dynamics and a number of computer programs model such unsteady-state operations. They are of paramount importance in the design and analysis of process control systems (see Process control). 

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