Auxiliary matrix

The latter one is governed by the auxiliary matrix equation (3.106) where... 

Construction of the Auxiliary Matrix for Evaluation of the Parameters of the Continued Fraction Starting from the Moments of the Hubbard Density of States. The Gordon PD Method is Applied. 

Calculations of relative amplitudes, quench vectors, and buffering of the species indicate that Per + and H2O2 are nonessential species. Since the enzymatic species are bound by a conservation constraint (their overall concentration is constant), the Jacobian matrix is singular and cannot be inverted. However, this can be circumvented by calculating the inverse of an auxiliary matrix [8], which provides concentration shifts with respect to only those species that are not involved in the conservation constraint. Leaving out the nonessential species, the concentration shifts with respect to admittable species (NAD, 0 , and O2) along with phase shifts relative to NAD (type X species) are summarized in table 11.12. 

Since the values of y can be calculated with a single integration of the system, in the function Mode lOdeExamplel, they are all evaluated in correspondence with the first experimental point, where ex is equal to 1. All the variables y are collected in an auxiliary BzzMatr lx Y and they are then recovered in correspondence with the value ex of the experiment. This device means there is no need to repeat the integration to get all the experimental points. The auxiliary matrix and other variables used at each iteration of the model are declared as static in order to prevent them from continually resizing. 

The auxiliary matrix and other variables that are used at each iteration of the model are declared as static in order to prevent them from continually... 

E((v-E(v))(v-E(v)) ), covariance matrix of adjustments for linear systems (E(v) = 0 if eo = 0) F, is pseudo-covariance matrix (10.3.31) for nonlinear systems covariance matrix (9.3.43) for estimate errors in measured variables for linear systems and assuming E(e) = 0 F is pseudo-covariance matrix (10.3.33) for nonlinear systems covariance matrix (9.3.19) of estimate errors in unmeasured observable variables for linear systems and assuming E(e) = 0 K is pseudo-covariance matrix (10.3.34) for nonlinear systems matrix Fj. in the case that all unmeasured variables are observable attention F is another matrix, viz. that introduced as auxiliary matrix in (9.2.26)... 

As before, we introduce the auxiliary matrix )(t), which is the solution of Equation (6.51) with the initial conditions... 

Absolute energy minimum principle. Yang developed two absolute energy minimum principles for first-principle linear-scaling electronic structure calculations. One is with a normalization constraint and the other without any constraint. The density matrix is represented by a set of nonorthogonal localized orbitals [i/] and an auxiliary matrix X which at the minimum becomes the (l)-inverse of the overlap matrix S of the localized orbitals. The number of localized orbitals is allowed to exceed the number of occupied orbitals. 

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