It should be clear that systems of linear algebraic equations can range in size from very small (fewer than five equations) to very large (several hundreds), depending on the number of components and complexity of operations. For example, a system consisting of four components being separated in a distillation column containing five stages yields a system of 20 material balance equations. Typically, the system of equations is rearranged into the following matrix form:
In this equation, [X] is the column matrix of n variables; [A], the n × n matrix of coefficients; and [B], a column matrix of n function values.
The Gauss elimination technique for solving this system of equations involves progressive elimination of variables from the equations such that at the end only a single linear equation is obtained in one variable. The value of that variable is then obtained and back-substituted progressively into the equations in reverse order of elimination to obtain the values of the rest of the variables that satisfy equation 4.15. For example, if the system consists of n equations in variables x1, x2,…, xn, then the first step is elimination of variable x1 from equations 2 to n using equation 1 to express x1 in terms of the rest of variables. The result is a system of n − 1 equations in n − 1 variables x2, x3,…, xn. Repeating this procedure then allows us to eliminate variables x2, x3, and so on, until only an equation in xn is left. The value of xn is calculated, and reversing the calculations, values of xn−1, xn−2,…, x1 are obtained [4].
Iterative procedures offer an alternative to elimination techniques. The Gauss-Seidel method involves assuming an initial solution by guessing the values for the variables. It is often convenient to assume that all the variables are 0. Based on this initial guess, the values of the variables are recalculated using the system of equations: x1 is calculated from the first equation, and its value is updated in the solution matrix; x2 is calculated from the second equation; and so on. The steps are repeated until the values converge for each variable [8]. The Gauss-Seidel method is likely to be more efficient than the elimination method for systems containing a very large number of equations or systems of equations with a sparse coefficient matrix, that is, where the majority of coefficients are zero [9].
Many sophisticated variations of the elimination and iteration techniques are available for the solution. One other solution technique involves matrix inversion and multiplication. The effectiveness of these and solution techniques is dependent on the nature of the system of equations. Certain techniques may work better in some situations, whereas it might be appropriate to use alternative techniques in other instances.
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