Gauss-Jordan Reduction Method

Discussion of Results The Gauss elimination method finds the interface temperatures as T, = 129.79°C, Tj = 129.77°C, and = 48.12°C. These values are quite predictable, because the heat transfer coefficient of steam and the heat conductivity of steel are very high. Therefore, the temperatures at steam-pipe interface and pipe-insulation interface are very close to the steam temperature. The main resistance to heat transfer is due to insulation. 

The values obtained from the function Gauss.m may be verified easily in MATLAB by using the original method of solution of the set of linear equations in matrix form, that is, T = 

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. 

We will apply the Gauss-Jordan procedure, without pivoting, to the setof Eqs. (2.I02) shown in Sec. 2..6.1 in order to observe the difference between the Gauss-Jordan and the Gauss method. Starting with the augmented matrix 

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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. 

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

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. 

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. 

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). 

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