Conditional convergence

The process of the exterior iterations continues to develop prior to the occurrence of the convergence conditions. 

Figure 7. An example of the potential profile with the metastable state (M) for which the convergence condition is disrupt.

At the same time, we see that practical application of this type of approximation requires the knowledge of the exact convergence conditions of these series. Surprisingly, the exact mathematical results here were obtained very recently (Passare and Tsikh, 2004) and the convergence theory still looks unfinished. 

In the case of the quadratic equation, the convergence condition for the "thermodynamic branch" series is simply positive discriminant (Passare and Tsikh, 2004). For kinetic polynomial (48) this discriminant is always positive for feasible values of parameters (see Equation (49)). This explains the convergence pattern for this series, in which the addition of new terms extended the convergence domain. 

If a boundary is natural the boundary condition is replaced by a convergence condition. For a natural repulsive boundary (i) one has Lx oo. If also L3 - oo there is no solution. If L3 < oo but L4 -+ oo, only a solution with J = 0 is possible, which is compatible with a reflecting boundary at the other end, but not with an absorbing boundary. If L3 < oo and L4 < oo no restriction results from the boundary, so that it is compatible with any boundary condition at the other end. 

Deciding where to stop is therefore almost completely a matter for the judgement of the user of a method. He must himself determine convergence conditions which are sufficient for his needs. These conditions must not be so slack that he stops too far from a minimum nor so strong that he wastes time computing quantities made up mostly of rounding and truncation errors. 

The convergence condition of the Taylor expansion (Equation 33) makes the series (Equation 35) convergent when... 

On the basis of the reactor model (Eq. 1 or 2), the estimator is designed by a geometric non-linear approach [5]. The approach follows a detectability property evaluation of the reactor motion to underlie the construction, tuning and convergence conditions of the estimator. 

Conditions 10.86 to 10.88 are frequently called the Courant-Friedrichs- Lewy (CFL) convergence conditions [58] and a is called the Courant number. 

The convergence condition stated above is difQcult to ascertain, since it requires eigenvalues of the Jacobian to be evaluated at the root. A relatively loose condition is to ensure that the eigenvalues of the Jacobian are less than unity in the interval where the expected root is located. Using this condition and the application of the Gershgorin theorem (BeU, 1965), the condition for the convergence of iterations can be ensured for the function g(x). 

Example 2.5 Application of the Convergence Condition Consider first-order reactions represented by... 

This statement is just a variant of well-known results applying to field theory, and in particular of a theorem established by Bergere and Lam.1 Let us note that, concerning our problems, this convergence condition at infinity depends on the space dimension d and that for special values of d it is not effective. 

The three-dimensional molecular electron density p(r) fulfills all the continuity, differentiability, and exponential convergence conditions specified for the function F(x) of the two-dimensional example. Consequently, a four-dimensional variant of the Alexandrov one-point compactification is feasible, and the three-dimensional space can be replaced by a three-dimensional sphere 5 embedded in a four-dimensional space R, using a one-to-one assignment of points r of space to the points r of the 3-sphere S. In addition, a single point, a formal north pole n of the sphere corresponds to all formal points of infinite displacement from the center of mass of the molecule. 

We neglect the energy balance. We chose the recycle as tear stream, so that the computational sequence is Mixer-Reactor-Separator. The tear stream has been cut explicitly in two parts, the streams 4 and 5. The convergence is obtained when the difference in component flow rates between the streams 4 and 5 is less than a prescribed tolerance. Let us denote the molar flow rates of the components. 4, fi, C in the stream 5 by, Fg, F(-. The convergence condition leads to the following algebraic equations ... 

Figure 6.12d The converged condition of the polarized calculation for the helium hydride ion, with the polarizing term turned fully on [ci/c2 set to 1.00 in cells G 24 and H 24],...

Comparison of Eq. [117] with the angle-constraint convergence condition leads to the following relation between the triangulation bond-stretch tolerance and the corresponding angle tolerance ... 

Now, the Euclidean norm of (r, - r,+i) is calculated to be 169.98. This does not sadsfy the convergence condition in the flowchart (Figure 7.A.1) so, t,- is updated with and the above calculations for exchangers 1 and 2 are repeated. The iterations are repeated until the Euclidean norm of (t,- - ) is less than a small value (5). 

Wolfe, P. (1969) Convergence conditions for ascent methods. SIAM Review,... 

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