Gauss Jordan

Extend the matrix triangularization procedure in Exercise 2-14 by the Gauss-Jordan procedure to obtain the fully diagonalized matrix solution set follows routinely. 

Gauss-Jordan elimination is a variation of the preceding method, which by continuation of the same procedures yields... 

If det C 0, C exists and can be found by matrix inversion (a modification of the Gauss-Jordan method), by writing C and 1 (the identity matrix) and then performing the same operations on each to transform C into I and, therefore, I into C". ... 

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. 

This set of linear equations can be solved by inspection, or, more formally, by Gauss-Jordan reduction of the augmented coefficient matrix ... 

The classification procedure developed by Madron is based on the conversion, into the canonical form, of the matrix associated with the linear or linearized plant model equations. First a composed matrix, involving unmeasured and measured variables and a vector of constants, is formed. Then a Gauss-Jordan elimination, used for pivoting the columns belonging to the unmeasured quantities, is accomplished. In the next phase, the procedure applies the elimination to a resulting submatrix which contains measured variables. By rearranging the rows and columns of the macro-matrix,... 

Because of aliasing, the total number of coefficients obtained should not be greater than N. We have a set of 2c linear equations for the 2c unknown coefficients. A number of standard methods are available for solving a set of linear equations. We used the Gauss-Jordan matrix reduction method. 

In this section we restrict considerations to an nxn nansingular matrix A. As shown in Section 1.1, the Gauss-Jordan elimination translates A into the identity matrix I. Selecting off-diagonal pivots we interchange some rows of I, and obtain a permutation matrix P instead, with exactly one element 1 in each row and in each column, all the other entries beeing zero. Matrix P is called permutation matrix, since the operation PA will interchange some rows of A. ... 

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... 

Solution of matrix equations bv Gauss-Jordan elimination... 

In Examples 1.1.2 and 1.1.3 we did not need the row interchange option of the program. This option is useful in pivoting, a practically indispensable auxiliary step in the Gauss-Jordan procedure, as will be discussed in the next section. Mhile the Gauss-Jordan procedure is a straightforward way of solving matrix equations, it is less efficient than some methods discussed later in this chapter. It is, however, almost as efficient as any other method to calculate the inverse of a matrix, the topics of our next section. 

Example 1.1.4 Inversion of a square matrix try Gauss-Jordan elimination. 

REH El. 1.1.4. INVERSION OF A NATRIX BY GAUSS-JORDAN ELIMINATION 104 REN HERGE N10... 

The use of the proposed iterative method reduced the error below 10" in all the cases, which means that a solution with at least 10 accurate digits has been obtained. The solution for x. obtained in this way has been compared to the solution calculated by different methods (i.e. Gauss-Jordan elimination). The results of the comparison verified the above conclusion. 

Scalar Fortran Coding of Gauss-Jordan Matrix Inverter... 

Any standard method of matrix inversion, such as the Gauss-Jordan method (N13), may be used to solve the equations. The coefficients in equations 4.11-4.14 may be used without serious error for most ordinary Portland cement clinkers in which the alite composition is not too different from that assumed here. As a byproduct of the calculation described in this section, and using the full compositions of the phases given in Table 1.2, one may calculate a mass balance table (Table 4.3) showing the distributions of all the oxide components among the phases. 

A multireaction version of Eq. (2.2-2) is obtainable by continuing the elimination above as well as below the pivotal elements this is the Gauss-Jordan algorithm of Section A.3. The final nonzero rows form a matrix a, here given by... 

If possible, bring 6i into the basis set 6 as the pivotal variable for row i and column i, by applying the modified Gauss-Jordan method described at the end of Section E.3 to the full A-matrix, including the last row and column. The requirements for such a move are LBAS(i) must be zero An must be positive (to ensure a descent of 5) Di and all resulting Dj must exceed ADTOL and every parameter must remain within the permitted region ofEq. (6.4-2). 

The transformed A-matrix quadratic expansion of S in terms of the set Be of current basis parameters. The submatrix A is the inverse of the current basis matrix. The basis determinant Aee is the product of the pivots An that were used in steps of Type 1, divided by the pivots A that were used in steps of Type 3. The use of strictly positive pivots for the Gauss-Jordan transformations ensures a positive definite submatrix A , hence a positive determinant Aee throughout the minimization. 

This chapter gives a brief summary of properties of linear algebraic equation systems, in elementary and partitioned form, and of certain elimination methods for their solution. Gauss-Jordan elimination, Gaussian elimination, LU factorization, and their use on partitioned arrays are described. Some software for computational linear algebra is pointed out, and references for further reading are given. 

Gauss-Jordan reduction is a straightforward elimination method that solves for an additional unknown Xk at each stage. An augmented array... 

For computation of a particular solution vector x, this method requires i(n -n)-l-n operations of multiplication or division, versus n —n) +n for the Gauss-Jordan method. Thus, the Gauss method takes about two-thirds as many operations. The computation of an inverse matrix takes 0(n ) operations of multiplication or division for either method. 

If you use the Gauss-Jordan method outlined in Sec. L. 1, you will find that the transformed augmented matrix for the set of equations above in the format of Eq. (L.4) will have only zeros in one row, indicating that one equation is not independent and hence redundant. (As explained in Appendix L, if the determinant of the augmented matrix were not equal to zero, the rank of the augmented matrix... 

In general there are two ways to solve Eq. (L.3) for Xi,. . ., x elimination techniques and iterative techniques. Both are easily executed by computer programs. In the pocket in the back cover of this book you will find a disk containing Fortran computer programs that can be used in solving sets of linear equations. We shall illustrate the Gauss-Jordan eliinination method. Other techniques can be found in texts on matrices, linear algebra, and numerical analysis. 

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