Auxiliary equation Lagrange

As indicated in Fig. 7, the next step after either an explicit or an implicit energy density functional orbit optimization procedure. For this purpose, one introduces the auxiliary functional Q[p(r) made up of the energy functional [p(r) 9 ]. plus the auxiliary conditions which must be imposed on the variational magnitudes. Notice that there are many ways of carrying out this variation, but that - in general - one obtains Euler-Lagrange equations by setting W[p(r) = 0. 

Let us consider now the energy density functional [/j(r,a) W] of Eq. (50). In this functional, p r, 5) stands for any one of the final densities generated from the initial density pg(r,s) coming from the orbit-generating wavefunction W. Clearly, the extremum of this functional is attained at the optimal density /> t(r,s). This density satisfies the Euler-Lagrange equation arising from the variation of the functional [/>(r,a) M] - subject to the normalization condition f d xp(r,s) = N -with respect to the one-particle density p(r, s). Explicitly, this amounts to varying the auxiliary functional... 

Equations can be formulated so that parameters of auxiliary elements can be set to zero. Also, the ON-resistance of switches can be set to zero turning them into ideal switches so that small time constants and thus a set of stiff model equations can be avoided. In the case residual sinks are used, which is similar to the use of Lagrange multipliers, the resulting mathematical model is a DAE system of index 2. 

If g is an Af-vector and z an W-vector then is an M-vector called vector of Lagrange multipliers. The problem then reads Find point (A -vector) z and M-vector as solutions of Eqs.(10.4.63). Thus generally, the problem is reformulated in terms of N+M nonlinear scalar equations in N+M variables, thus in the components of z and the auxiliary vector X. ... 

---