Gauss-Jordan decomposition

Since the last reaction is a linear combination of the first two (sum), it can be easily proved that the rank remains unchanged at 2. So to conclude, the concentrations of all components in this network can be expressed in terms of two, say H2 and Freon 12, and the first two reactions form an independent reaction set. In case of more complicated networks it may be difficult to determine the independent reactions by observation alone. In this case the Gauss-Jordan decomposition leads to a set of independent reactions (see, e.g., Amundson, Mathematical Methods in Chemical Engineering—Matrices and Their Application, Prentice-Hall International, New York, 1966). 

According to Scales (1985) the best way to solve Equation 5.12b is by performing a Cholesky factorization of the Hessian matrix. One may also perform a Gauss-Jordan elimination method (Press et al., 1992). An excellent user-oriented presentation of solution methods is provided by Lawson and Hanson (1974). We prefer to perform an eigenvalue decomposition as discussed in Chapter 8. 

Since det(A) =0 if and only if A is singular, it provides a convenient way of checking singularity. Determinants have traditionally been used also for solving matrix equations (ref. 10), but both the Gauss-Jordan method and the Gaussian elimination are much more efficient. The determinant itself can easily be calculated by LU decomposition. For the decomposed matrix (1.45)... 

A special property of solving a matrix equation in this way is that the LU decomposition does not involve the right-hand side vector b, in contrast both to the Gauss-Jordan method and to the Gaussian elimination. This is... 

Typical procedures to solve the OLS problem are Gaussian elimination and Gauss-Jordan elimination. More efficient solutions are based on decomposition of the X matrix by algorithms, such as LU decomposition. Householder reduction, or singular value decomposition (SVD). One of the most powerful methods, SVD, is outlined as follows (cf. Section 5.2 and Biased Parameter Estimations PCR and PLS Section). 

Cramer s rule is usually sufficient for solving two equations in two unknowns or three equations in three unknowns. However, for larger sets of equations, other solution procedures are preferred, such as Gauss-Jordan reduction and the Gauss-Seidel method. But in most cases, the best method is LU decomposition, in which the coeffi-... 

The differentiation of Eq. (10) with respect to each G, gives sets of equations, which are called the normal equations of the linear least-squares problem. These normal equations can be solved by Gauss-Jordan elimination. However, in many cases the normal equations are very close to singular and a zero pivot element may be encountered. In such cases instead of using the normal equations, Eq. (10) can be solved by singular value decomposition. 

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