Equations irreducible sets

To improve the formatting of the equations that represent a plant, many commercial codes partition the equations into groups of irreducible sets of equations, that is, those that have to be solved simultaneously. If a plant is represented by thousands of equations, the overall time consumed in their solution via either a GRG or SQP algorithm is reduced by partitioning and rearranging the order of the equations with the result indicated in Figure 15.6. Organization of the set of equations into irreducible sets can be carried out by the use of permutation matrices or by one of... 

B. How the Irreducible Sets of Equations (Those to be Solved Simultaneously)... 

B. How the Irreducible Sets of Equations (Those to be Solved Simultaneously) Correspond to the Maximal Loops in the Adjacency Matrix and are Invariant of the Output Set... 

The nonzero elements of a column, j, of the matrix P indicate the loops that are fed information by the equations of loop j, and the nonzero elements of any row, i, indicate the loops that feed information to loop i. The ordering procedure for the irreducible sets of equatins (maximal loops) is as follows ... 

From Fig. E5.6 you can see that it is not necessary to solve all 10 equations simultaneously. The system of equations can be broken up into lower-order subsystems, some of which can be comprised of individual equations or small groups of equations. These groups are associated with the so-called irreducible sets of equations. By inspection you can establish the following... 

An example of such an irreducible set is given in the flow network of Figure 18.3, assumed for simplicity to carry liquid at a constant temperature. This network can hardly be described as complicated, but it cannot be reduced further using either parallel or serial transformations, and we shall need to solve an implicit equation in order to find the intermediate pressure, P2, and the flows, Wc, Wjj and W24, as will now be demonstrated. 

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. 

In tearing, the objective is to wind up with less computation time required to solve the torn system compared with the time required to solve the entire block of equations simultaneously. However, the criteria for evaluating the effectiveness of the tearing are by no means so well defined as those for partitioning, where the objective is clearly to obtain the smallest possible subsystems of irreducible equations. There is no general method for determining the time needed to effect a solution of a set of equations it is necessary to consider the particular equations involved. Any feasible method of tearing, then, must be based on criteria that are related to the solution time. Some of the more obvious criteria are ... 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

Let us summarize the general procedure we would follow for an arbitrary reaction system. For the batch or PFTR we write S first-order differential equations describing each of the S species. We then eliminate as many equations as possible by finding S — R suitable stoichiometric relations among species to obtain R irreducible equations. Sometimes some of these are uncoupled from others so that we may need to solve smaller sets simultaneously. 

In the second place, the Hamiltonian operators which occur and commute with all Or belong to the totally symmetric irreducible representation T1 (see Appendix A. 10-3) and integrals over them dT vanish unless T = T (see eqn (8-4.5)). Thus, in carrying out an approximate solution of the electronic Schrodinger equation, changing to a set of basis functions which belong to the irreducible representations will allow us, by inspection, to put many of the integrals which occur equal to zero. There will also, because of this, be an... 

Hence we have the matrix equation RS = W, so that the matrices (9.44) multiply the same way the symmetry operations do and form a representation of the point group. The functions Fl,F2,...,Fn are said to form a basis for the representation (9.44), which consists of the matrices that describe how these functions transform upon application of the symmetry operators. Any member of the set Fv...,Fn is said to belong to the representation (9.44). We denote the representation (9.44) by TF it may be reducible or irreducible. 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

Equation (9.31) implies that two nonequivalent irreducible representations cannot have the same set of characters. 

This means that in the set of matrices constituting any one irreducible representation any set of corresponding matrix elements, one from each matrix, behaves as the components of a vector in /i-dimensional space such that all these vectors are mutually orthogonal, and each is normalized so that the square of its length equals hllh This-interpretation of 4.3-1 will perhaps be more obvious if we take 4.3-1 apart into three simpler equations, each of which is contained within it. We shall omit the explicit designation of complex conjugates for simplicity, but it should be remembered that they must be used... 

PROOF. As for rule 1, a complete proof will not be given we can, however, easily prove that the number of classes sets an upper limit on the number of irreducible representations. We can combine 4.3-6 and 4.3-7 into one equation, namely,... 

For the p-shell we do not need these additional relationships, since from the set of equations (15.71)—(15.74), (15.82) and (15.83) we can work out expressions for all the irreducible tensorial products. Directly from (15.71) we obtain (hereinafter a letter over the equality sign shows what shell this particular equation is valid for)... 

It can be shown that the set of matrices US(P) generated in this way constitute an irreducible representation of the group The set of functions sk hence form a basis for this representation, and the degeneracy in the level E implied by equation (12) is termed the permutational degeneracy it has no physical significance. 

Finally it should be noted that if the molecular system has any spatial symmetry, i.e. if there is a point group 0 whose operations St all commute with H, then each function sk in equation (7) must be replaced by a set which forms a basis for an irreducible representation A of 0 ... 

The set of matrices Us Pr) in equation (41) form a reducible representation of the group which is reduced into its irreducible components A by the coefficients a above. If we denote the irreducible representations of iAn and 2 by 17 Is- iVl and Z)(/1>, respectively, then this reduction can be written symbolically as... 

Since a second-rank cartesian tensor Tap transforms in the same way as the set of products uaVfj, it can also be expressed in terms of a scalar (which is the trace T,y(y), a vector (the three components of the antisymmetric tensor (1 /2 ) Tap — Tpaj), and a second-rank spherical tensor (the five components of the traceless, symmetric tensor, (I /2)(Ta/= + Tpa) - (1/3)J2Taa). The explicit irreducible spherical tensor components can be obtained from equations (5.114) to (5.118) simply by replacing u vp by T,/ . These results are collected in table 5.2. It often happens that these three spherical tensors with k = 0, 1 and 2 occur in real, physical situations. In any given situation, one or more of them may vanish for example, all the components of T1 are zero if the tensor is symmetric, Yap = Tpa. A well-known example of a second-rank spherical tensor is the electric quadrupole moment. Its components are defined by... 

Note that there is more than one way of representing the dipolar interaction in irreducible tensors, as discussed in appendix 8.2. Equation (9.41) can be compared with the coupling choice used in our analysis of the H/ spectrum described in chapter 11. It is partly a matter of choice, but more especially a matter of the basis set used in the analysis. 

The smoothness of algebraic matrix groups is a property not shared by all closed sets in /c". To see what it means, take fc = fc and let 5 fc" be an arbitrary irreducible closed set. Let s be a point in S corresponding to the maximal ideal J in k[S]. If S is smooth, n si k = O si /J us) has fc-dimension equal to the dimension of S. (This would in general be called smoothness at s.) If S is defined by equations fj = 0, the generators and relations for OUS] show that S is smooth at s iff the matrix of partial derivatives (dfj/dXi)(s) has rank n — dim V. Over the real or complex field this is the standard Jacobian criterion for the solutions of the system (f = 0) to form a C or analytic submanifold near s. For S to be smooth means then that it has no cusps or self-crossings or other singularities . 

A method for precedence ordering so as to partition a model into a sequence of smaller models containing sets of irreducible equations (equations that have to be solved simultaneously as illustrated in Fig. 5.6). 

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