Iteration, monotone

The equations are solved by the Rabitz iterative monotonous algorithm [24], We have used the improvement proposed in Ref. [46]. At each iteration k, the field is given by where is calculated by Eq. (16). 

The subsequent representations are probably reliable within the range of data used (always less broad than 200° to 600°K), but they are only approximations outside that range. The functions are, however, always monotonic in temperature, to provide appropriate corrections when iterative programs choose temperature excursions outside the range of data. 

We consider penalized operator equations approximating variational inequalities. For equations with strongly monotonous operators we construct an iterative method, prove convergence of solutions, and obtain error estimates. 

Kovtunenko V.A. (1994b) An iteration penalty method for variational inequalities with strongly monotonous operators. Siberian Math. J. 35 (4), 735-738. 

For nonquadratic but monotonic surfaces, the Newton-Raphson minimization method can be applied near a minimum in an iterative way [24]. 

S. HeikkilS and V. Lakshmikantham, Monotone iterative Techniques for Discontinuous Nonlinear Differential Equations (1994)... 

The solution of these dynamic nonlinear differential equations is considerably more complex than the previous systems considered. In particular, stable solution methods are based on physically realistic multiphase flow functions that have the following properties relative permeability functions are non-negative, monotoni-cally increasing with their respective saturation, and are zero at vanishing saturations, and capillary pressure is monotonically increasing with respect to the saturation of the non-wetting phase. It is necessary that any iterative scheme for estimating the multiphase flow functions retain these characteristics at each step. 

Monotone iteration methods for solving coupled systems of nonlinear boundary value problems (with K. Zygourakis). Comput. Chem. Eng. 7, 183-193 (1983). 

Remark 4 The termination criterion in step 6 is based on obtaining a larger value in the primal problems for consecutive iterations. However, there is no theoretical reason why this should be the criterion for termination since the primal problems by their definition do not need to satisfy any monotonicity property. (i.e they do not need to be nonincreasing). Based on the above, this termination criterion can be viewed only as a heuristic and may result in premature termination of the OA/ER/AP. Therefore, the OA/ER/AP can fail to identify the global solution of the MINLP problem (6.33). 

Since dF/dV is always negative, the F vs. V relation is monotonic, and this makes Newton s method (App. E), a rapidly converging iteration procedure, well suited to solution for V. Newton s method here gives... 

Iteration and convergence method explicit equations Monotone sequences and secant method Newton- Raphson Free ion molali-ties by difference Newton- Raphson conti nued fraction Newton- Raphson Newton-Raphson conti nued fraction conti nued fraction for anions only conti nued fraction conti nued fraction conti nued fraction brute force... 

Both local and global searches stop either when the computational effort employed exceeds the maximum used in previous iterations and a constraint violation has been identified or when the specified maximum effort has been employed unsuccessfully. This ensures monotonically increasing search effort as the outer approximation improves and the design converges. 

In applying Newton s method it is crucial that f (.r) is monotonic in the interval [a, 6]. If this is not the case, an iterative solution of Eq. (D.6) may not converge to the correct value of. tq. In this case, Newton s method may, however, still work if the intial value x ) is selected by an educated guess that is, x needs to be chosen sufficiently close to xq where / (x) is still monotonic. 

The recursive computation usually converges in a small number of iterations due to the smooth and monotonic nature of the function/(v)-... 

The value of Lt = 382.6 Ibmole/hr computed in the second iteration is considerably higher than the value of 327.7 Ibmole/hr from the initial iteration. More importantly, the quantities absorbed in the first and second iterations are 162.7 and 217.6 Ibmole/hr, respectively, the second value being 34% higher than the first. To obtain converged results, additional iterations are conducted in the same manner as the second iteration. The results for L are as follows, where the difference between successive values of is monotonically reduced. 

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