Condition coefficient matrix

Furthermore, the implementation of the Gauss-Newton method also incorporated the use of the pseudo-inverse method to avoid instabilities caused by the ill-conditioning of matrix A as discussed in Chapter 8. In reservoir simulation this may occur for example when a parameter zone is outside the drainage radius of a well and is therefore not observable from the well data. Most importantly, in order to realize substantial savings in computation time, the sequential computation of the sensitivity coefficients discussed in detail in Section 10.3.1 was implemented. Finally, the numerical integration procedure that was used was a fully implicit one to ensure stability and convergence over a wide range of parameter estimates. 

An equivalent statement is that no row of the coefficient matrix (j8) can be formed as a linear combination of the other rows. Since the matrix s determinant is nonzero when and only when this statement is true, we need only evaluate the determinant of (/ ) to demonstrate that a new basis B is valid. In practice, this test can be accomplished using a linear algebra package, or implicitly by testing for error conditions produced while inverting the matrix, since a square matrix has an inverse if and only if its determinant is not zero. 

The results presented above were discussed in terms of the special case of elementary reactions. However, if we relax the condition that the coefficients vfai and uTai must be integers, (5.1) is applicable to nearly all chemical reactions occurring in practical applications. In this general case, the element conservation constraints are no longer applicable. Nevertheless, all of the results presented thus far can be expressed in terms of the reaction coefficient matrix T, defined as before by... 

In Section 5.1, we have seen (Fig. 5.2) that the molar concentration vector c can be transformed using the SVD of the reaction coefficient matrix T into a vector c that has Nr reacting components cr and N conserved components cc.35 In the limit of equilibrium chemistry, the behavior of the Nr reacting scalars will be dominated by the transformed chemical source term S. 36 On the other hand, the behavior of the N conserved scalars will depend on the turbulent flow field and the inlet and initial conditions for the flow domain. However, they will be independent of the chemical reactions, which greatly simplifies the mathematical description. 

In words, this condition states that plJ) for j e Nmf + 1,..., Ain (except j = k) must be a linear mixture of p for i e 1,..., AW with the same coefficient matrix B needed for the mixture-fraction vector. Hereinafter, a mixture-fraction basis that satisfies (5.104) will be referred to as a linear-mixture basis. 

The reaction coefficient matrix T for this case is rank one. Thus, a linear transformation can be found that generates one reacting scalar and two conserved scalars. Moreover, if the flow system has only two inlet streams, and the initial conditions are a linear mixture of the... 

The new coefficient matrix is symmetric as M lA can be written as M 1/2AM 1/Z. Preconditioning aims to produce a more clustered eigenvalue structure for M A and/or lower condition number than for A to improve the relevant convergence ratio however, preconditioning also adds to the computational effort by requiring that a linear system involving M (namely, Mz = r) be solved at every step. Thus, it is essential for efficiency of the method that M be factored very rapidly in relation to the original A. This can be achieved, for example, if M is a sparse component of the dense A. Whereas the solution of an n X n dense linear system requires order of 3 operations, the work for sparse systems can be as low as order n.13-14... 

Under the multi-term condition, the lower level sub-problem of the hybrid tabu search needs to decide the network routes for every period. It refers to a linear program whose coefficient matrix becomes almost block diagonal per each period and expands rapidly with the number of terms as known from the foregoing statement. 

The form of the SCEP treatment will vary in certain aspects depending upon whether it is employed to carry out a Cl, CC or Moller-Plesset (MP) perturbation theory calculation. However, the differences are modest and the same quantities appear in one place or another. For convenience we utilize here the MP perturbation theory version of SCEP as formulated by Pulay and Saebo [30, 31] for their local correlation treatment. The (Hylleraas) variation condition on the first-order coefficient matrix, C = CP, may be written in the form... 

An alternative to the methods described above can be used if the coefficient matrix is diagonalizable. Consider, once again, the matrix differential equation and its associated initial condition... 

There are several sub-groups of MWR methods, according to the particular choices for the weighting function Wi z) employed. The performance of the resulting MWR is to a certain extent tied to the properties of the resulting coefficient matrix. To enable an efficient solution process it is desired that the coefficient matrix is symmetric, positive definite and characterized by a small condition number. At the same time the work needed to assemble the coefficient vaiues shouid be minimized. 

The determinant of the coefficient matrix vanishes, A2 = 0, thus this condition is satisfied for A = 0. 

Therefore, we see that the boundary conditions at the interface lead to a set of four linear algebraic equations for the constants A, C), B2, D2. The condition for existence of a nontrivial solution of this set of algebraic equations is that the determinant of the coefficient matrix must equal to zero. This condition leads to a complicated algebraic equation relating the dimensionless growth-rate coefficient a to the dimensionless wave number a for specified values of the fluid viscosities, the fluid densities, and the interfacial tension. As usual, stability is determined by the sign of the real part of a. 

The condition for nontrivial solutions is that the determinate of the coefficient matrix must be equal to zero. After some algebraic manipulation (which can be done with one of the computer algebra programs), this leads to... 

Figure 5.7. When the initial and inlet conditions admit a linear-mixture basis, the molar concentration vector c of length K can be partitioned by a linear transformation into three parts (p, a reaction-progress vector of length Nr 4, a mixture-fraction vector of length N f and 0, a null vector of length K — Nr — AW- The linear transformation matrix E depends on the reference concentration vector c and the reaction coefficient matrix T.

A is a coefficient matrix that is designed to transform between solutions that obey arbitrary boundary conditions and those which obey the desired boundary conditions. A and S can be regarded as unknowns in equation (A3.11.72) and equation (A3.11.73). This leads to the following expression for S ... 

Matrix algebra is employed to solve Ny + 2 systems of linear algebraic equations. However, flie problem is simplified when the coefficient matrix Aj is tridiagonal. Under these conditions, the Thomas algorithm (Carnahan et al., 1969, pp. 441 -442) provides an efficient solution for all unknown molar densities at Zk+ along the grid line where yj is constant. This procedure is repeated - - 2 times to calculate all unknown molar densities in the flow cross section at ZAr+i. 

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