GAUSS-JORDAN REDUCTION

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

Find Qpfc, the absolutely largest current element in column k that lies in a row not yet selected for pivoting. If Qpfc is negligible, skip to step 4 otherwise, accept it as the pivot for stage k and go to step 2. 

Eliminate from all rows but p by transforming the array as follows  

The use of the largest available pivot element, Upk, in column k limits the growth of rounding error by ensuring that the multipliers ouk/ocpk) in Eq. (A.3-2) all have magnitude 1 or less. 

This algorithm takes min(m, n) stages. The total number of pivots accepted is the rank, r, of the equation system.The final array can be decoded by associating each coefficient column with the corresponding variable. Rearranging the array to place the pivotal rows and columns in the order of selection, one can write the results in the partitioned form 

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

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-... 

To avoid this waste of time and money, and also to use those samples that have the largest spread of variations not only in those variables whose values are known but also in those variables that we may not even know exist but that have an effect on the spectrum, what is needed is a method of selecting samples that has certain characteristics. Those characteristics are as follows (a) It is based on measurements of the optical data only, and (b) it selects the samples that show the most differences in the spectra. An algorithm based on Gauss-Jordan reduction has been developed to overcome this problem [7], and here we present an approach based on the Mahalanobis distance concept. 

Table 2.3 illustrates how the number of operations required by Cramer s rule increases as the value of n increases. Forn = 3, a total of 51 multiplications and divisions are needed. However, when n = 10, this number climbs to 359,251,210. For this reason, Cramer s rule is rarely used for systems with n > 3, The Gauss elimination, Gauss-Jordan reduction, and Gauss-Seidel methods, to be described in the next three sections of this chapter, are much more efficient methods of solution of linear equations than Cramer s rule. 

The Gauss-Jordan reduction method is an extension of the Gauss elimination method. It reduces a set of n equations from its canonical form of... 

The Gauss-Jordan reduction method applies the same series of elementary operations that are used by the Gauss elimination method. It applies these operations both below and above the diagonal in order to reduce all the off-diagonal elements of the matrix to zero. In addition, it converts the elements on the diagonal to unity. 

The Gauss-Jordan reduction procedure applied to the ( ) x (n + I) augmented matrix can be given in a three-part mathematical formula for the initialization, nonnalization, and reduction steps as shown below ... 

Gauss-Jordan Reduction with Matrix inversion... 

This simply states that the inverse of A is equal to L. This has veiy important implications in numerical method.s because it shows that the Gauss-Jordan reduction method is essentially a matrix inversion algorithm Eq. (2.136), when rearranged, clearly shows that the application of the reduction operation L on the identity matrix yields the inverse of A ... 

Example 2.2 demonstrates the use of the Gauss-Jordan reduction method for the solution of simultaneous linear algebraic equations. 

Example 2.2 Solution of a Steam Distribution System Using the Gauss-jordan Reduction Method for Simultaneous Linear Algebraic Equations. Figure E2.2 represents the steam distribution system of a chemical plant. The material and energy balances of this system are given below ... 

Solves a set of simultaneous linear algebraic equations that model the steam distribution system of a chemical plant using the Gauss-Jordan Reduction method (Jordan.m). 

Gauss-Jordan Reduction method for solution of simultaneous linear algebraic equations. 

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