Inner and outer iterations

The optimization an MCSCF wave function is carried out as a sequence of inner and outer iterations. The outer iterations are those discussed in Sections 12.3 and 12.4 - the individual iterations of the second-order methods carried out to converge the wave function. The inner iterations are those carried out to solve the linear equations (12.5.1) or the eigenvalue equations (12.5.2). The inner and outer iterations are also referred to as micro and macro iterations. 

The iteration process continues through the inner and outer loops. 

The iteration consists in solving Eqs. (4.80) and (4.81) with a starting value of B. Subsequently the following expressions are determined which enable one to compare the exact result (left hand side of Eq. (4.73), denoted by A) and the approximate result (right hand side of Eq. (4.73), denoted by B). J is the choice of the number of mesh points between the inner and outer radius, e.g. 350. 

Fig.2 Test of the rotational RKR method. Solid and open dots refer to inner and outer turning points for J=0-30 in steps of 5. Residuals are riven for (a) Lennard-Jones (b) Morse (c) no iterative refinement. (Taken irom ref.5 with permission).

Modification of Describing Equations. In the simple K-value and enthalpy models just described, there are Ng+6 parameters, a, A, B, C, D, E and F, which have characteristics that make them excellent choices for the iteration variables of an outer iteration loop. This is in fact the essence of the inside-out concept. Its success, however, rests on the ability to transform and rearrange the describing equations to properly accommodate these variables. The result should be an efficient and well-behaved inner iteration loop in which values of the primitive variables - now regarded as dependent variables - are calculated. When the inside loop is converged, the actual K-value and enthalpy models can be employed to calculate new values of the simple model parameters. 

DEW P. The calculational scheme here is shown in Fig. 12.13. We read and store T and yk, along with appropriate constants. Since we can calculate neither the d> nor the yk, all values of each are set equal to unity. Values of P are found from the Antoine equation, and Eqs. (12.29) and (12.27) are then solved for P and xk. Evaluation of y now allows recalculation of P by Eq. (12.29). With this rather good estimate of P, we evaluate and enter an inner iteration loop that converges on values for xk and (yj. Subsequent recalculation of P by Eq. (12.29) leads to the outer iteration loop that establishes the final value of P. Since the xk calculated within the inner loop are not constrained to sum to... 

For unsteady flows the system of non-linear equations are linearized in the iteration process within each time step, since all the solvers are limited to linear systems. The iterative process is thus performed on two different levels. The solver iterations are performed on provisional linear systems with fixed coefficients and source terms until convergence. Then, the system coefficients and sources are updated based on the last provisional solution and a new linearized system is solved. This process is continued until the non-linear system is converged, meaning that two subsequent linear systems give the same solution within the accuracy of a prescribed criterion. A standard notation used for the different iterations within one time step is that the coefficient and source matrices are updated in the outer iterations, whereas the inner iterations are performed on provisionally linear systems with fixed coefficients. On each outer iteration, the equations solved are on the form ... 

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