Solution of Linear Algebraic Equations

Linear systems of algebraic equations occur often in science and engineering as exact or approximate formulations of various problems. Such systems have appeared many times in this book. Exact linear problems appear in Chapter 2, and iterative linearization schemes are used in Chapters 6 and 7. 

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. 

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The solution of linear algebraic equations by this method is based on the following steps ... 

Multigrid methods have proven to be powerful algorithms for the solution of linear algebraic equations. They are to be considered as a combination of different techniques allowing specific weaknesses of iterative solvers to be overcome. For this reason, most state-of-the-art commercial CFD solvers offer the multigrid capability. 

The inverse matrix also can be employed in the solution of linear algebraic equations,... 

The subject of linear equations is best described in terms of concepts associated with linear algebra and matrix theory. The reader is referred to Amundson (1966) for details. We present here only the basic definitions and results that are important for the solution of linear algebraic equations. Consider m equations in n unknowns x, X2,. ..,x given by... 

In the procedure which is analogous to the Gauss-Seidel solution of linear algebraic equations, the most recent value of each variable is used at each point in the calculational procedure. The Gauss-Seidel type of iteration differs from the Jacobi type of iteration in that instead of computing each element of Xk on the basis of an assumed vector Xfc 1 as indicated by Eq. (15-23), the following procedure is employed... 

The catalytic reaction mechanisms and corresponding kinetic models can be classified into linear and non-linear models. These terms were introduced by Temkin [2]. For linear mechanisms every reaction involves the participation of only one molecule of the intermediate substance. Therefore the rate of each step depends linearly on an intermediate concentration. Using the principle of pseudo-steady-state concentrations (see equation (7)) we can easily find the solution of linear algebraic equations that corresponds to the linear mechanism and then obtain the values of the pseudo-steady-state (or steady-state)... 

Stagewise calculations require the simultaneous solution of material and energy balances with equilibrium relationships. It was demonstrated in Example 1.1 that the design of a simple extraction system reduces to the solution of linear algebraic equations if (1) no energy balances are needed and (2) the equilibrium relationship is linear. 

Notice that all the nonzero entries in the matrix are grouped around the diagonal elements. In fact, we have no more than two elements on either side of the diagonal for this problem. In order to solve for the unknown time level temperatures, this matrix must be inverted. Verj efficient algorithms have been developed for the solution of linear algebraic equations with band type matrices (Von Rosenberg, 1969). 

Another example requiring the solution of linear algebraic equations comes from the analysis of complex reaction systems that have monomolecular kinetics. Fig. 2.2 considers a chemical reaction between the three species, whose concentrations are designated by Fj, Fj, taking place in a batch reactor. 

In developing systematic methods for the solution of linear algebraic equations and the evaluation of eigenvalues and eigenvectors of linear systems, we will make extensive use of matrix-vector notation. For this reason, and for the benefit of the reader, a review of selected matrix and vector operations is given in the next section. 

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